Subgroups — C₆₇×#25877 #30705128

Order 17152.

1254 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3500	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2861	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2038	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2037	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1967	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1962	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1793	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1792	C₂	1	2	yes	maximal, complemented
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
14	#548×C₆₇	—	2	2	no	—
15	#546×C₆₇	—	2	2	no	—
16	#546×C₆₇	—	2	4	no	—
17	#536×C₆₇	—	2	1	no	—
18	#536×C₆₇	C₂²	1	1	yes	characteristic
19	#535×C₆₇	—	2	1	no	—
20	#535×C₆₇	C₂²	1	1	yes	characteristic
21	#532×C₆₇	—	2	1	no	—
22	#532×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#124×C₆₇	—	2	1	no	—
25	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
27	#491×C₆₇	—	2	1	no	—
28	#490×C₆₇	—	2	1	no	—
29	#488×C₆₇	—	2	1	no	—
30	#488×C₆₇	—	2	1	no	—
31	#479×C₆₇	—	2	2	no	—
32	#478×C₆₇	—	2	2	no	—
33	#428×C₆₇	C₂²	1	2	yes	complemented
34	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	C₂×#104×C₆₇	C₂²	1	2	yes	—
37	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
38	#417×C₆₇	C₂²	1	2	yes	—
39	#417×C₆₇	C₂²	1	2	yes	—
40	#417×C₆₇	C₂²	1	2	yes	complemented
41	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
42	C₂×#100×C₆₇	C₂²	1	2	yes	—
43	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#391×C₆₇	C₂²	1	2	yes	complemented
45	#390×C₆₇	C₂²	1	2	yes	—
46	#388×C₆₇	—	2	4	no	—
47	C₄³×C₆₇	C₂²	1	1	yes	characteristic, complemented
48	#339×C₆₇	C₂²	1	2	yes	complemented
49	2₋¹⁺⁴×C₆₇	—	2	4	no	—
50	2₋¹⁺⁴×C₆₇	—	4	2	no	—
51	H(5,𝔽₂)×C₆₇	—	2	4	no	—
52	H(5,𝔽₂)×C₆₇	—	4	2	no	—
53	C₂×#41×C₆₇	—	2	1	no	—
54	C₂×#41×C₆₇	—	2	1	no	—
55	C₂×#41×C₆₇	—	2	1	no	—
56	C₂×#41×C₆₇	—	2	1	no	—
57	C₂×#41×C₆₇	—	2	2	no	—
58	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₆₇	—	2	1	no	—
62	C₂²×Q₈×C₆₇	—	2	1	no	—
63	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₆₇	—	2	2	no	—
65	C₂²×D₄×C₆₇	—	2	1	no	—
66	C₂²×D₄×C₆₇	—	2	1	no	—
67	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₆₇	—	2	1	no	—
71	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
72	#127×C₆₇	—	2	1	no	—
73	#127×C₆₇	D₄	1	1	yes	characteristic
74	#125×C₆₇	—	2	2	no	—
75	#125×C₆₇	—	2	1	no	—
76	#125×C₆₇	—	2	1	no	—
77	#124×C₆₇	—	2	4	no	—
78	#124×C₆₇	—	2	1	no	—
79	#124×C₆₇	—	2	1	no	—
80	#124×C₆₇	—	2	1	no	—
81	#124×C₆₇	—	2	1	no	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	D₄	1	1	yes	characteristic
84	#124×C₆₇	D₄	1	1	yes	characteristic
85	#123×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	2	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	2	no	—
89	#121×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	4	no	—
91	#121×C₆₇	—	2	2	no	—
92	#121×C₆₇	—	2	2	no	—
93	#120×C₆₇	—	2	4	no	—
94	#120×C₆₇	—	2	2	no	—
95	C₄×Q₈×C₆₇	D₄	1	2	yes	—
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	2	no	—
98	C₄×D₄×C₆₇	—	2	2	no	—
99	C₄×D₄×C₆₇	D₄	1	2	yes	—
100	#117×C₆₇	—	2	1	no	—
101	#117×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
106	C₂×C₂²⋊C₄×C₆₇	D₄	1	4	yes	—
107	C₂×C₄²×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	—	2	2	no	—
109	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	#107×C₆₇	—	2	2	no	—
112	#106×C₆₇	—	2	2	no	—
113	C₄⋊C₈×C₆₇	—	2	2	no	—
114	#104×C₆₇	—	2	4	no	—
115	#104×C₆₇	—	4	2	no	—
116	#104×C₆₇	—	4	2	no	—
117	#103×C₆₇	—	2	2	no	—
118	#103×C₆₇	—	2	2	no	—
119	#103×C₆₇	—	2	2	no	—
120	#102×C₆₇	—	2	2	no	—
121	#102×C₆₇	—	2	2	no	—
122	#102×C₆₇	—	2	2	no	—
123	#101×C₆₇	—	4	2	no	—
124	#100×C₆₇	—	4	2	no	—
125	#97×C₆₇	—	2	1	no	—
126	C₄×C₈×C₆₇	—	2	1	no	—
127	#95×C₆₇	—	2	4	no	—
128	C₂⁴×C₆₇	—	2	2	no	—
129	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
130	#41×C₆₇	—	2	2	no	—
131	#41×C₆₇	—	2	2	no	—
132	#41×C₆₇	—	2	2	no	—
133	#41×C₆₇	—	2	2	no	—
134	#41×C₆₇	—	2	2	no	—
135	#41×C₆₇	—	4	4	no	—
136	#41×C₆₇	—	4	8	no	—
137	#41×C₆₇	—	4	4	no	—
138	#41×C₆₇	—	4	2	no	—
139	#41×C₆₇	—	4	2	no	—
140	#41×C₆₇	—	4	2	no	—
141	#41×C₆₇	—	4	2	no	—
142	#41×C₆₇	#41	1	4	yes	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂×Q₈×C₆₇	—	2	2	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	4	4	no	—
149	C₂×Q₈×C₆₇	—	4	2	no	—
150	C₂×Q₈×C₆₇	—	4	2	no	—
151	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	2	2	no	—
158	C₂×D₄×C₆₇	—	2	1	no	—
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	2	no	—
162	C₂×D₄×C₆₇	—	2	1	no	—
163	C₂×D₄×C₆₇	—	2	1	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	—	4	4	no	—
167	C₂×D₄×C₆₇	—	4	8	no	—
168	C₂×D₄×C₆₇	—	4	2	no	—
169	C₂×D₄×C₆₇	—	4	2	no	—
170	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	2	no	—
178	C₂²×C₄×C₆₇	—	2	1	no	—
179	C₂²×C₄×C₆₇	—	2	1	no	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	2	no	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	M₁₆×C₆₇	—	2	2	no	—
186	M₁₆×C₆₇	—	2	2	no	—
187	M₁₆×C₆₇	—	2	2	no	—
188	M₁₆×C₆₇	—	2	2	no	—
189	C₂×C₈×C₆₇	—	2	2	no	—
190	C₂×C₈×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	2	no	—
196	C₄⋊C₄×C₆₇	—	4	2	no	—
197	C₄⋊C₄×C₆₇	—	4	2	no	—
198	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	1	no	—
202	C₂²⋊C₄×C₆₇	—	2	2	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	2	no	—
206	C₂²⋊C₄×C₆₇	—	2	2	no	—
207	C₂²⋊C₄×C₆₇	—	4	8	no	—
208	C₄²×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	—	2	2	no	—
211	C₄²×C₆₇	—	2	2	no	—
212	C₄²×C₆₇	—	4	4	no	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₂³×C₆₇	—	2	1	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	4	4	no	—
222	C₂³×C₆₇	—	4	4	no	—
223	C₂³×C₆₇	—	8	4	no	—
224	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₆₇	—	2	1	no	—
226	Q₈×C₆₇	—	2	2	no	—
227	Q₈×C₆₇	—	2	2	no	—
228	Q₈×C₆₇	—	4	4	no	—
229	Q₈×C₆₇	—	4	1	no	—
230	Q₈×C₆₇	—	4	1	no	—
231	Q₈×C₆₇	—	4	2	no	—
232	Q₈×C₆₇	#124	1	2	yes	—
233	D₄×C₆₇	—	2	4	no	—
234	D₄×C₆₇	—	2	1	no	—
235	D₄×C₆₇	—	2	2	no	—
236	D₄×C₆₇	—	2	2	no	—
237	D₄×C₆₇	—	4	4	no	—
238	D₄×C₆₇	—	4	4	no	—
239	D₄×C₆₇	—	4	4	no	—
240	D₄×C₆₇	—	4	2	no	—
241	D₄×C₆₇	—	4	2	no	—
242	D₄×C₆₇	—	4	1	no	—
243	D₄×C₆₇	—	4	1	no	—
244	D₄×C₆₇	—	4	2	no	—
245	D₄×C₆₇	—	8	2	no	—
246	D₄×C₆₇	—	8	4	no	—
247	D₄×C₆₇	#124	1	2	yes	—
248	C₂×C₄×C₆₇	—	2	1	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	4	2	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	2	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	2	no	—
262	C₂×C₄×C₆₇	—	4	1	no	—
263	C₂×C₄×C₆₇	—	4	1	no	—
264	C₂×C₄×C₆₇	—	4	1	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	4	no	—
267	C₂×C₄×C₆₇	—	4	2	no	—
268	C₂×C₄×C₆₇	—	8	2	no	—
269	C₂×C₄×C₆₇	#120	1	2	yes	—
270	C₂×C₄×C₆₇	#120	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
274	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
275	C₈×C₆₇	—	4	2	no	—
276	C₈×C₆₇	—	4	2	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	2	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	1	no	—
281	C₂²×C₆₇	—	4	1	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	2	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	8	4	no	—
287	C₂²×C₆₇	—	16	4	no	—
288	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
289	C₂²×C₆₇	#546	1	2	yes	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	2	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	4	4	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	4	1	no	—
296	C₄×C₆₇	—	4	1	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	8	2	no	—
299	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
300	C₄×C₆₇	#535	1	1	yes	characteristic
301	C₄×C₆₇	#546	1	2	yes	—
302	#25877	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
303	C₂×C₆₇	—	2	2	no	—
304	C₂×C₆₇	—	4	2	no	—
305	C₂×C₆₇	—	8	1	no	—
306	C₂×C₆₇	—	8	2	no	—
307	C₂×C₆₇	—	8	1	no	—
308	C₂×C₆₇	#2435	1	1	yes	characteristic
309	C₂×C₆₇	#3018	1	2	yes	—
310	#3500	C₂×C₆₇	1	1	yes	characteristic
311	#3142	C₂×C₆₇	1	1	yes	characteristic, complemented
312	#3080	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#2861	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2038	C₂×C₆₇	1	2	yes	complemented
315	#2037	C₂×C₆₇	1	2	yes	complemented
316	#1967	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#1962	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#1793	C₂×C₆₇	1	2	yes	complemented
319	#1792	C₂×C₆₇	1	2	yes	complemented
320	C₄×#104	C₂×C₆₇	1	1	yes	characteristic, complemented
321	C₆₇	#25877	1	1	yes	characteristic, direct product factor, Sylow
322	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
324	#548	—	2	2	no	—
325	#546	—	2	2	no	—
326	#546	—	2	4	no	—
327	#536	—	2	1	no	—
328	#536	C₂²×C₆₇	1	1	yes	characteristic
329	#535	—	2	1	no	—
330	#535	C₂²×C₆₇	1	1	yes	characteristic
331	#532	—	2	1	no	—
332	#532	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
334	C₂×#124	—	2	1	no	—
335	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
336	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
337	#491	—	2	1	no	—
338	#490	—	2	1	no	—
339	#488	—	2	1	no	—
340	#488	—	2	1	no	—
341	#479	—	2	2	no	—
342	#478	—	2	2	no	—
343	#428	C₂²×C₆₇	1	2	yes	complemented
344	#423	C₂²×C₆₇	1	1	yes	characteristic, complemented
345	C₂×#104	C₂²×C₆₇	1	2	yes	—
346	C₂×#104	C₂²×C₆₇	1	2	yes	—
347	C₂×#104	C₂²×C₆₇	1	2	yes	complemented
348	#417	C₂²×C₆₇	1	2	yes	—
349	#417	C₂²×C₆₇	1	2	yes	—
350	#417	C₂²×C₆₇	1	2	yes	complemented
351	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
352	C₂×#100	C₂²×C₆₇	1	2	yes	—
353	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
354	#391	C₂²×C₆₇	1	2	yes	complemented
355	#390	C₂²×C₆₇	1	2	yes	—
356	#388	—	2	4	no	—
357	C₄³	C₂²×C₆₇	1	1	yes	characteristic, complemented
358	#339	C₂²×C₆₇	1	2	yes	complemented
359	2₋¹⁺⁴	—	2	4	no	—
360	2₋¹⁺⁴	—	4	2	no	—
361	H(5,𝔽₂)	—	2	4	no	—
362	H(5,𝔽₂)	—	4	2	no	—
363	C₂×#41	—	2	1	no	—
364	C₂×#41	—	2	1	no	—
365	C₂×#41	—	2	1	no	—
366	C₂×#41	—	2	1	no	—
367	C₂×#41	—	2	2	no	—
368	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
369	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
370	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
371	C₂²×Q₈	—	2	1	no	—
372	C₂²×Q₈	—	2	1	no	—
373	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	—	2	1	no	—
377	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
379	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
380	C₄⋊Q₈	—	2	1	no	—
381	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
382	#127	—	2	1	no	—
383	#127	D₄×C₆₇	1	1	yes	characteristic
384	#125	—	2	2	no	—
385	#125	—	2	1	no	—
386	#125	—	2	1	no	—
387	#124	—	2	4	no	—
388	#124	—	2	1	no	—
389	#124	—	2	1	no	—
390	#124	—	2	1	no	—
391	#124	—	2	1	no	—
392	#124	—	2	2	no	—
393	#124	D₄×C₆₇	1	1	yes	characteristic
394	#124	D₄×C₆₇	1	1	yes	characteristic
395	#123	—	2	4	no	—
396	#123	—	2	2	no	—
397	C₂²⋊Q₈	—	2	4	no	—
398	C₂²⋊Q₈	—	2	2	no	—
399	#121	—	2	4	no	—
400	#121	—	2	4	no	—
401	#121	—	2	2	no	—
402	#121	—	2	2	no	—
403	#120	—	2	4	no	—
404	#120	—	2	2	no	—
405	C₄×Q₈	D₄×C₆₇	1	2	yes	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	D₄×C₆₇	1	2	yes	—
410	#117	—	2	1	no	—
411	#117	C₂³×C₆₇	1	1	yes	characteristic
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₂²⋊C₄	—	2	4	no	—
415	C₂×C₂²⋊C₄	—	2	4	no	—
416	C₂×C₂²⋊C₄	D₄×C₆₇	1	4	yes	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	—	2	2	no	—
419	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	#107	—	2	2	no	—
422	#106	—	2	2	no	—
423	C₄⋊C₈	—	2	2	no	—
424	#104	—	2	4	no	—
425	#104	—	4	2	no	—
426	#104	—	4	2	no	—
427	#103	—	2	2	no	—
428	#103	—	2	2	no	—
429	#103	—	2	2	no	—
430	#102	—	2	2	no	—
431	#102	—	2	2	no	—
432	#102	—	2	2	no	—
433	#101	—	4	2	no	—
434	#100	—	4	2	no	—
435	#97	—	2	1	no	—
436	C₄×C₈	—	2	1	no	—
437	#95	—	2	4	no	—
438	C₂⁴	—	2	2	no	—
439	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
440	#41	—	2	2	no	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	#41	—	2	2	no	—
444	#41	—	2	2	no	—
445	#41	—	4	4	no	—
446	#41	—	4	8	no	—
447	#41	—	4	4	no	—
448	#41	—	4	2	no	—
449	#41	—	4	2	no	—
450	#41	—	4	2	no	—
451	#41	—	4	2	no	—
452	#41	#41×C₆₇	1	4	yes	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	4	4	no	—
459	C₂×Q₈	—	4	2	no	—
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	4	4	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	4	2	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
481	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
482	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	M₁₆	—	2	2	no	—
496	M₁₆	—	2	2	no	—
497	M₁₆	—	2	2	no	—
498	M₁₆	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	4	2	no	—
507	C₄⋊C₄	—	4	2	no	—
508	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	4	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	4	4	no	—
523	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
524	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
525	C₄²	C₂×D₄×C₆₇	1	2	yes	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	4	4	no	—
532	C₂³	—	4	4	no	—
533	C₂³	—	8	4	no	—
534	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
535	Q₈	—	2	1	no	—
536	Q₈	—	2	2	no	—
537	Q₈	—	2	2	no	—
538	Q₈	—	4	4	no	—
539	Q₈	—	4	1	no	—
540	Q₈	—	4	1	no	—
541	Q₈	—	4	2	no	—
542	Q₈	#124×C₆₇	1	2	yes	—
543	D₄	—	2	4	no	—
544	D₄	—	2	1	no	—
545	D₄	—	2	2	no	—
546	D₄	—	2	2	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	1	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	2	no	—
555	D₄	—	8	2	no	—
556	D₄	—	8	4	no	—
557	D₄	#124×C₆₇	1	2	yes	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	4	no	—
568	C₂×C₄	—	4	4	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	8	2	no	—
579	C₂×C₄	#120×C₆₇	1	2	yes	—
580	C₂×C₄	#120×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
584	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
585	C₈	—	4	2	no	—
586	C₈	—	4	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	8	4	no	—
597	C₂²	—	16	4	no	—
598	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
599	C₂²	#546×C₆₇	1	2	yes	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	8	2	no	—
609	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
610	C₄	#535×C₆₇	1	1	yes	characteristic
611	C₄	#546×C₆₇	1	2	yes	—
612	C₂	—	2	2	no	—
613	C₂	—	4	2	no	—
614	C₂	—	8	2	no	—
615	C₂	—	8	1	no	—
616	C₂	—	8	1	no	—
617	C₂	C₆₇×#2435	1	1	yes	characteristic
618	C₂	C₆₇×#3018	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#3142	C₂	semidirect
3. C₆₇×#3080	C₂	semidirect
4. C₆₇×#2861	C₂	semidirect
5. C₆₇×#2038	C₂	semidirect
6. C₆₇×#2037	C₂	semidirect
7. C₆₇×#1967	C₂	semidirect
8. C₆₇×#1962	C₂	semidirect
9. C₆₇×#1793	C₂	semidirect
10. C₆₇×#1792	C₂	semidirect
11. C₄×#104×C₆₇	C₂	semidirect
23. C₂×#125×C₆₇	C₂²	semidirect
33. #428×C₆₇	C₂²	semidirect
34. #423×C₆₇	C₂²	semidirect
37. C₂×#104×C₆₇	C₂²	semidirect
40. #417×C₆₇	C₂²	semidirect
41. C₂×#101×C₆₇	C₂²	semidirect
43. C₄×M₁₆×C₆₇	C₂²	semidirect
44. #391×C₆₇	C₂²	semidirect
47. C₄³×C₆₇	C₂²	semidirect
48. #339×C₆₇	C₂²	semidirect
302. #25877	C₆₇	direct (N × H)
311. #3142	C₂×C₆₇	semidirect
312. #3080	C₂×C₆₇	semidirect
313. #2861	C₂×C₆₇	semidirect
314. #2038	C₂×C₆₇	semidirect
315. #2037	C₂×C₆₇	semidirect
316. #1967	C₂×C₆₇	semidirect
317. #1962	C₂×C₆₇	semidirect
318. #1793	C₂×C₆₇	semidirect
319. #1792	C₂×C₆₇	semidirect
320. C₄×#104	C₂×C₆₇	semidirect
321. C₆₇	#25877	direct (N × H)
333. C₂×#125	C₂²×C₆₇	semidirect
343. #428	C₂²×C₆₇	semidirect
344. #423	C₂²×C₆₇	semidirect
347. C₂×#104	C₂²×C₆₇	semidirect
350. #417	C₂²×C₆₇	semidirect
351. C₂×#101	C₂²×C₆₇	semidirect
353. C₄×M₁₆	C₂²×C₆₇	semidirect
354. #391	C₂²×C₆₇	semidirect
357. C₄³	C₂²×C₆₇	semidirect
358. #339	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3500	2
C₆₇×#3142	2
C₆₇×#3080	2
C₆₇×#2861	2
C₆₇×#2038	2
C₆₇×#2037	2
C₆₇×#1967	2
C₆₇×#1962	2
C₆₇×#1793	2
C₆₇×#1792	2
C₄×#104×C₆₇	2
#25877	67

Subgroups — C67×#25877 #30705128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25877 #30705128