Subgroups — C₆₇×#22468 #30701719

Order 17152.

1316 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2898	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1908	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1882	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1793	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
13	#578×C₆₇	—	2	4	no	—
14	#578×C₆₇	C₄	1	8	yes	complemented
15	C₂×#137×C₆₇	—	2	2	no	—
16	C₂×#137×C₆₇	C₂²	1	2	yes	—
17	C₂×#135×C₆₇	—	2	1	no	—
18	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₆₇	—	2	1	no	—
20	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₆₇	C₂²	1	2	yes	—
22	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
23	#520×C₆₇	C₂²	1	2	yes	—
24	#443×C₆₇	—	2	4	no	—
25	#440×C₆₇	—	2	2	no	—
26	C₄×Q₁₆×C₆₇	—	2	1	no	—
27	C₄×D₈×C₆₇	—	2	1	no	—
28	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#104×C₆₇	C₂²	1	2	yes	—
32	#419×C₆₇	—	2	2	no	—
33	#419×C₆₇	C₂²	1	2	yes	—
34	#417×C₆₇	—	2	2	no	—
35	#417×C₆₇	C₂²	1	2	yes	—
36	#417×C₆₇	C₂²	1	2	yes	—
37	#416×C₆₇	—	2	2	no	—
38	#416×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#103×C₆₇	—	2	2	no	—
40	C₂×#103×C₆₇	C₂²	1	2	yes	—
41	C₂×#101×C₆₇	C₂²	1	2	yes	—
42	#405×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#391×C₆₇	C₂²	1	2	yes	—
45	#343×C₆₇	C₂²	1	2	yes	—
46	#341×C₆₇	C₂²	1	1	yes	characteristic
47	#339×C₆₇	C₂²	1	1	yes	characteristic
48	2₋¹⁺⁴×C₆₇	—	2	4	no	—
49	2₋¹⁺⁴×C₆₇	—	2	4	no	—
50	2₋¹⁺⁴×C₆₇	—	4	2	no	—
51	2₋¹⁺⁴×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	4	2	no	—
55	C₂×#41×C₆₇	C₂³	1	2	yes	—
56	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	—	2	2	no	—
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	#137×C₆₇	—	2	4	no	—
61	#137×C₆₇	—	2	16	no	—
62	#137×C₆₇	C₂×C₄	1	8	yes	complemented
63	#135×C₆₇	—	2	8	no	—
64	#135×C₆₇	C₂×C₄	1	4	yes	—
65	#135×C₆₇	D₄	1	4	yes	—
66	C₂×Q₁₆×C₆₇	—	2	2	no	—
67	C₂×Q₁₆×C₆₇	—	2	8	no	—
68	C₂×Q₁₆×C₆₇	—	2	1	no	—
69	C₂×Q₁₆×C₆₇	—	2	1	no	—
70	C₂×Q₁₆×C₆₇	—	2	2	no	—
71	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₆₇	—	2	2	no	—
74	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
75	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
76	#131×C₆₇	—	2	2	no	—
77	#131×C₆₇	—	2	4	no	—
78	#131×C₆₇	—	4	2	no	—
79	#131×C₆₇	C₂×C₄	1	4	yes	complemented
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	—	2	2	no	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₂²×C₈×C₆₇	—	2	2	no	—
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	4	2	no	—
92	#117×C₆₇	—	2	2	no	—
93	#117×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	#108×C₆₇	—	2	2	no	—
98	#107×C₆₇	—	2	1	no	—
99	#107×C₆₇	—	2	1	no	—
100	#104×C₆₇	—	2	4	no	—
101	#104×C₆₇	—	2	2	no	—
102	#103×C₆₇	—	2	2	no	—
103	#103×C₆₇	—	2	2	no	—
104	#103×C₆₇	—	2	2	no	—
105	#103×C₆₇	—	2	2	no	—
106	#103×C₆₇	—	2	2	no	—
107	#103×C₆₇	—	2	2	no	—
108	#103×C₆₇	D₄	1	1	yes	characteristic
109	#103×C₆₇	D₄	1	1	yes	characteristic
110	#102×C₆₇	—	2	2	no	—
111	#102×C₆₇	—	2	2	no	—
112	#102×C₆₇	D₄	1	1	yes	characteristic
113	#102×C₆₇	D₄	1	1	yes	characteristic
114	#101×C₆₇	—	4	2	no	—
115	#97×C₆₇	—	2	1	no	—
116	C₄×C₈×C₆₇	—	2	2	no	—
117	C₄×C₈×C₆₇	—	2	1	no	—
118	#95×C₆₇	—	4	2	no	—
119	#41×C₆₇	—	2	1	no	—
120	#41×C₆₇	—	2	2	no	—
121	#41×C₆₇	—	2	2	no	—
122	#41×C₆₇	—	2	1	no	—
123	#41×C₆₇	—	2	8	no	—
124	#41×C₆₇	—	2	8	no	—
125	#41×C₆₇	—	4	8	no	—
126	#41×C₆₇	—	4	4	no	—
127	#41×C₆₇	—	4	4	no	—
128	#41×C₆₇	—	4	8	no	—
129	#41×C₆₇	C₂²×C₄	1	4	yes	—
130	#41×C₆₇	C₂²×C₄	1	2	yes	—
131	#41×C₆₇	C₂×D₄	1	4	yes	—
132	#41×C₆₇	#41	1	2	yes	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	8	no	—
135	C₂×Q₈×C₆₇	—	2	8	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	4	2	no	—
141	C₂×Q₈×C₆₇	—	4	4	no	—
142	C₂×Q₈×C₆₇	—	4	4	no	—
143	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	4	2	no	—
149	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	2	no	—
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₆₇	—	2	8	no	—
161	Q₁₆×C₆₇	—	2	4	no	—
162	Q₁₆×C₆₇	—	2	4	no	—
163	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
164	SD₁₆×C₆₇	—	2	8	no	—
165	SD₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
166	D₈×C₆₇	C₂²⋊C₄	1	4	yes	—
167	M₁₆×C₆₇	—	2	1	no	—
168	M₁₆×C₆₇	—	2	4	no	—
169	M₁₆×C₆₇	—	2	2	no	—
170	M₁₆×C₆₇	—	2	1	no	—
171	M₁₆×C₆₇	—	2	1	no	—
172	M₁₆×C₆₇	—	2	2	no	—
173	M₁₆×C₆₇	—	4	4	no	—
174	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	1	no	—
178	C₂×C₈×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	2	1	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	4	4	no	—
182	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
183	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	4	2	no	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₂²⋊C₄×C₆₇	—	2	1	no	—
194	C₂²⋊C₄×C₆₇	—	4	2	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	1	no	—
201	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
202	Q₈×C₆₇	—	2	1	no	—
203	Q₈×C₆₇	—	2	2	no	—
204	Q₈×C₆₇	—	2	4	no	—
205	Q₈×C₆₇	—	2	4	no	—
206	Q₈×C₆₇	—	4	8	no	—
207	Q₈×C₆₇	—	4	2	no	—
208	Q₈×C₆₇	—	4	2	no	—
209	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
210	Q₈×C₆₇	C₄×D₄	1	2	yes	—
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	2	4	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	2	1	no	—
216	D₄×C₆₇	—	2	2	no	—
217	D₄×C₆₇	—	4	8	no	—
218	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
219	D₄×C₆₇	C₄×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	4	8	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	8	2	no	—
238	C₂×C₄×C₆₇	—	16	2	no	—
239	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
245	C₈×C₆₇	—	2	2	no	—
246	C₈×C₆₇	—	2	1	no	—
247	C₈×C₆₇	—	2	1	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₈×C₆₇	—	4	1	no	—
250	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
251	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
252	C₂²×C₆₇	—	2	1	no	—
253	C₂²×C₆₇	—	2	1	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	1	no	—
260	C₂²×C₆₇	#518	1	1	yes	characteristic
261	C₂²×C₆₇	D₄²	1	1	yes	characteristic
262	C₂²×C₆₇	#546	1	1	yes	characteristic
263	C₄×C₆₇	—	2	2	no	—
264	C₄×C₆₇	—	2	4	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	4	no	—
267	C₄×C₆₇	—	4	1	no	—
268	C₄×C₆₇	—	4	1	no	—
269	C₄×C₆₇	—	8	2	no	—
270	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
271	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
272	C₄×C₆₇	D₄²	1	1	yes	characteristic
273	C₄×C₆₇	#546	1	1	yes	characteristic
274	#22468	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	2	1	no	—
277	C₂×C₆₇	—	4	2	no	—
278	C₂×C₆₇	—	4	4	no	—
279	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₆₇	#3290	1	1	yes	characteristic
281	C₂×C₆₇	#3293	1	1	yes	characteristic
282	C₂×#578	C₂×C₆₇	1	1	yes	characteristic
283	C₂×#443	C₂×C₆₇	1	1	yes	characteristic, complemented
284	#2951	C₂×C₆₇	1	1	yes	characteristic, complemented
285	#2898	C₂×C₆₇	1	2	yes	complemented
286	#1979	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1969	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#1908	C₂×C₆₇	1	2	yes	complemented
289	#1882	C₂×C₆₇	1	2	yes	complemented
290	#1816	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#1793	C₂×C₆₇	1	2	yes	complemented
292	#1764	C₂×C₆₇	1	1	yes	characteristic, complemented
293	C₆₇	#22468	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
295	#578	—	2	4	no	—
296	#578	C₄×C₆₇	1	8	yes	complemented
297	C₂×#137	—	2	2	no	—
298	C₂×#137	C₂²×C₆₇	1	2	yes	—
299	C₂×#135	—	2	1	no	—
300	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
301	C₂²×Q₁₆	—	2	1	no	—
302	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#131	C₂²×C₆₇	1	2	yes	—
304	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
305	#520	C₂²×C₆₇	1	2	yes	—
306	#443	—	2	4	no	—
307	#440	—	2	2	no	—
308	C₄×Q₁₆	—	2	1	no	—
309	C₄×D₈	—	2	1	no	—
310	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×#104	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#104	C₂²×C₆₇	1	2	yes	—
314	#419	—	2	2	no	—
315	#419	C₂²×C₆₇	1	2	yes	—
316	#417	—	2	2	no	—
317	#417	C₂²×C₆₇	1	2	yes	—
318	#417	C₂²×C₆₇	1	2	yes	—
319	#416	—	2	2	no	—
320	#416	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#103	—	2	2	no	—
322	C₂×#103	C₂²×C₆₇	1	2	yes	—
323	C₂×#101	C₂²×C₆₇	1	2	yes	—
324	#405	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
326	#391	C₂²×C₆₇	1	2	yes	—
327	#343	C₂²×C₆₇	1	2	yes	—
328	#341	C₂²×C₆₇	1	1	yes	characteristic
329	#339	C₂²×C₆₇	1	1	yes	characteristic
330	2₋¹⁺⁴	—	2	4	no	—
331	2₋¹⁺⁴	—	2	4	no	—
332	2₋¹⁺⁴	—	4	2	no	—
333	2₋¹⁺⁴	C₂×C₄×C₆₇	1	8	yes	complemented
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	4	2	no	—
337	C₂×#41	C₂³×C₆₇	1	2	yes	—
338	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
339	C₂²×Q₈	—	2	2	no	—
340	C₂²×Q₈	—	2	2	no	—
341	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
342	#137	—	2	4	no	—
343	#137	—	2	16	no	—
344	#137	C₂×C₄×C₆₇	1	8	yes	complemented
345	#135	—	2	8	no	—
346	#135	C₂×C₄×C₆₇	1	4	yes	—
347	#135	D₄×C₆₇	1	4	yes	—
348	C₂×Q₁₆	—	2	2	no	—
349	C₂×Q₁₆	—	2	8	no	—
350	C₂×Q₁₆	—	2	1	no	—
351	C₂×Q₁₆	—	2	1	no	—
352	C₂×Q₁₆	—	2	2	no	—
353	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
354	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
355	C₂×SD₁₆	—	2	2	no	—
356	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
357	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
358	#131	—	2	2	no	—
359	#131	—	2	4	no	—
360	#131	—	4	2	no	—
361	#131	C₂×C₄×C₆₇	1	4	yes	complemented
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	—	2	2	no	—
364	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
365	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
367	C₂²×C₈	—	2	1	no	—
368	C₂²×C₈	—	2	2	no	—
369	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
371	C₄×Q₈	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	4	2	no	—
374	#117	—	2	2	no	—
375	#117	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
379	#108	—	2	2	no	—
380	#107	—	2	1	no	—
381	#107	—	2	1	no	—
382	#104	—	2	4	no	—
383	#104	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#103	—	2	2	no	—
388	#103	—	2	2	no	—
389	#103	—	2	2	no	—
390	#103	D₄×C₆₇	1	1	yes	characteristic
391	#103	D₄×C₆₇	1	1	yes	characteristic
392	#102	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	D₄×C₆₇	1	1	yes	characteristic
395	#102	D₄×C₆₇	1	1	yes	characteristic
396	#101	—	4	2	no	—
397	#97	—	2	1	no	—
398	C₄×C₈	—	2	2	no	—
399	C₄×C₈	—	2	1	no	—
400	#95	—	4	2	no	—
401	#41	—	2	1	no	—
402	#41	—	2	2	no	—
403	#41	—	2	2	no	—
404	#41	—	2	1	no	—
405	#41	—	2	8	no	—
406	#41	—	2	8	no	—
407	#41	—	4	8	no	—
408	#41	—	4	4	no	—
409	#41	—	4	4	no	—
410	#41	—	4	8	no	—
411	#41	C₂²×C₄×C₆₇	1	4	yes	—
412	#41	C₂²×C₄×C₆₇	1	2	yes	—
413	#41	C₂×D₄×C₆₇	1	4	yes	—
414	#41	#41×C₆₇	1	2	yes	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	8	no	—
417	C₂×Q₈	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	4	no	—
424	C₂×Q₈	—	4	4	no	—
425	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
426	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
427	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
432	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
442	Q₁₆	—	2	8	no	—
443	Q₁₆	—	2	4	no	—
444	Q₁₆	—	2	4	no	—
445	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
446	SD₁₆	—	2	8	no	—
447	SD₁₆	C₂²⋊C₄×C₆₇	1	8	yes	—
448	D₈	C₂²⋊C₄×C₆₇	1	4	yes	—
449	M₁₆	—	2	1	no	—
450	M₁₆	—	2	4	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	1	no	—
453	M₁₆	—	2	1	no	—
454	M₁₆	—	2	2	no	—
455	M₁₆	—	4	4	no	—
456	M₁₆	C₂²×C₄×C₆₇	1	2	yes	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	4	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	4	4	no	—
464	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
465	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
467	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
469	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	4	2	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	4	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	1	no	—
483	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
484	Q₈	—	2	1	no	—
485	Q₈	—	2	2	no	—
486	Q₈	—	2	4	no	—
487	Q₈	—	2	4	no	—
488	Q₈	—	4	8	no	—
489	Q₈	—	4	2	no	—
490	Q₈	—	4	2	no	—
491	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
492	Q₈	C₄×D₄×C₆₇	1	2	yes	—
493	D₄	—	2	2	no	—
494	D₄	—	2	4	no	—
495	D₄	—	2	2	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	1	no	—
498	D₄	—	2	2	no	—
499	D₄	—	4	8	no	—
500	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
501	D₄	C₄×D₄×C₆₇	1	2	yes	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	8	2	no	—
520	C₂×C₄	—	16	2	no	—
521	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
526	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
527	C₈	—	2	2	no	—
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	4	2	no	—
531	C₈	—	4	1	no	—
532	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
533	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	1	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	1	no	—
542	C₂²	#518×C₆₇	1	1	yes	characteristic
543	C₂²	D₄²×C₆₇	1	1	yes	characteristic
544	C₂²	#546×C₆₇	1	1	yes	characteristic
545	C₄	—	2	2	no	—
546	C₄	—	2	4	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	4	no	—
551	C₄	—	8	2	no	—
552	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
553	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
554	C₄	D₄²×C₆₇	1	1	yes	characteristic
555	C₄	#546×C₆₇	1	1	yes	characteristic
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	4	2	no	—
559	C₂	—	4	4	no	—
560	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
561	C₂	C₆₇×#3290	1	1	yes	characteristic
562	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#578×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₆₇	C₂	semidirect
3. C₆₇×#2951	C₂	semidirect
4. C₆₇×#2898	C₂	semidirect
5. C₆₇×#1979	C₂	semidirect
6. C₆₇×#1969	C₂	semidirect
7. C₆₇×#1908	C₂	semidirect
8. C₆₇×#1882	C₂	semidirect
9. C₆₇×#1816	C₂	semidirect
10. C₆₇×#1793	C₂	semidirect
11. C₆₇×#1764	C₂	semidirect
14. #578×C₆₇	C₄	semidirect
51. 2₋¹⁺⁴×C₆₇	C₂×C₄	semidirect
62. #137×C₆₇	C₂×C₄	semidirect
71. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
79. #131×C₆₇	C₂×C₄	semidirect
274. #22468	C₆₇	direct (N × H)
283. C₂×#443	C₂×C₆₇	semidirect
284. #2951	C₂×C₆₇	semidirect
285. #2898	C₂×C₆₇	semidirect
286. #1979	C₂×C₆₇	semidirect
287. #1969	C₂×C₆₇	semidirect
288. #1908	C₂×C₆₇	semidirect
289. #1882	C₂×C₆₇	semidirect
290. #1816	C₂×C₆₇	semidirect
291. #1793	C₂×C₆₇	semidirect
292. #1764	C₂×C₆₇	semidirect
293. C₆₇	#22468	direct (N × H)
296. #578	C₄×C₆₇	semidirect
333. 2₋¹⁺⁴	C₂×C₄×C₆₇	semidirect
344. #137	C₂×C₄×C₆₇	semidirect
353. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
361. #131	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#578×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2951	2
C₆₇×#2898	2
C₆₇×#1979	2
C₆₇×#1969	2
C₆₇×#1908	2
C₆₇×#1882	2
C₆₇×#1816	2
C₆₇×#1793	2
C₆₇×#1764	2
#22468	67

Subgroups — C67×#22468 #30701719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22468 #30701719