Subgroups — C₆₇×#25924 #30705175

Order 17152.

1138 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3138	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3038	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2027	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1883	C₂	1	4	yes	maximal, complemented
9	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
11	Q₈²×C₆₇	—	2	1	no	—
12	#557×C₆₇	—	2	1	no	—
13	#554×C₆₇	—	2	1	no	—
14	#554×C₆₇	—	2	2	no	—
15	D₄×Q₈×C₆₇	—	2	2	no	—
16	D₄×Q₈×C₆₇	—	2	1	no	—
17	#544×C₆₇	—	2	2	no	—
18	#544×C₆₇	C₂²	1	2	yes	—
19	#533×C₆₇	—	2	1	no	—
20	#533×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#124×C₆₇	—	2	2	no	—
25	#520×C₆₇	C₂²	1	2	yes	—
26	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
27	#490×C₆₇	—	2	2	no	—
28	#486×C₆₇	—	2	1	no	—
29	#486×C₆₇	—	2	1	no	—
30	#460×C₆₇	—	2	1	no	—
31	#460×C₆₇	—	2	1	no	—
32	#459×C₆₇	—	2	1	no	—
33	#459×C₆₇	—	2	1	no	—
34	#423×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#104×C₆₇	C₂²	1	2	yes	—
36	#418×C₆₇	C₂²	1	2	yes	—
37	#418×C₆₇	C₂²	1	4	yes	—
38	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	#390×C₆₇	—	2	2	no	—
40	#390×C₆₇	C₂²	1	4	yes	—
41	C₄³×C₆₇	C₂²	1	1	yes	characteristic
42	#343×C₆₇	C₂²	1	4	yes	—
43	#339×C₆₇	C₂²	1	4	yes	—
44	#339×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
51	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
52	C₂×SD₁₆×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
55	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	—	2	2	no	—
58	C₄⋊Q₈×C₆₇	—	2	4	no	—
59	C₄⋊Q₈×C₆₇	—	2	2	no	—
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	—	2	1	no	—
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	C₄⋊Q₈×C₆₇	—	2	1	no	—
64	C₄⋊Q₈×C₆₇	—	2	1	no	—
65	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
67	#127×C₆₇	—	2	1	no	—
68	#127×C₆₇	—	2	1	no	—
69	#127×C₆₇	—	2	1	no	—
70	#127×C₆₇	—	2	1	no	—
71	#125×C₆₇	—	2	4	no	—
72	#125×C₆₇	—	2	2	no	—
73	#125×C₆₇	—	2	2	no	—
74	#125×C₆₇	D₄	1	2	yes	—
75	#124×C₆₇	—	4	8	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	2	no	—
78	C₂²⋊Q₈×C₆₇	—	2	4	no	—
79	C₂²⋊Q₈×C₆₇	—	2	4	no	—
80	C₂²⋊Q₈×C₆₇	—	2	1	no	—
81	C₂²⋊Q₈×C₆₇	—	2	1	no	—
82	C₂²⋊Q₈×C₆₇	—	2	4	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	—	2	2	no	—
85	C₄×Q₈×C₆₇	—	2	2	no	—
86	C₄×Q₈×C₆₇	—	2	1	no	—
87	C₄×Q₈×C₆₇	—	2	1	no	—
88	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	2	1	no	—
92	C₄×D₄×C₆₇	—	2	1	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
95	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
96	#117×C₆₇	—	2	2	no	—
97	#117×C₆₇	—	2	1	no	—
98	#117×C₆₇	C₂³	1	1	yes	characteristic
99	#117×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
105	C₂×C₄²×C₆₇	—	2	1	no	—
106	C₂×C₄²×C₆₇	—	2	1	no	—
107	C₂×C₄²×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₆₇	—	2	1	no	—
111	C₄⋊C₈×C₆₇	—	2	1	no	—
112	#104×C₆₇	—	2	4	no	—
113	#102×C₆₇	—	2	2	no	—
114	#102×C₆₇	—	2	2	no	—
115	#102×C₆₇	—	2	4	no	—
116	#102×C₆₇	—	2	4	no	—
117	#97×C₆₇	—	2	1	no	—
118	C₄×C₈×C₆₇	—	2	1	no	—
119	C₂⁴×C₆₇	—	2	2	no	—
120	#41×C₆₇	—	2	2	no	—
121	#41×C₆₇	#41	1	4	yes	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	—	2	1	no	—
124	C₂×Q₈×C₆₇	—	2	2	no	—
125	C₂×Q₈×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	4	4	no	—
128	C₂×Q₈×C₆₇	—	8	4	no	—
129	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	4	no	—
134	C₂×D₄×C₆₇	—	2	4	no	—
135	C₂×D₄×C₆₇	—	4	4	no	—
136	C₂×D₄×C₆₇	—	4	4	no	—
137	C₂×D₄×C₆₇	—	4	1	no	—
138	C₂×D₄×C₆₇	—	4	1	no	—
139	C₂×D₄×C₆₇	—	4	1	no	—
140	C₂×D₄×C₆₇	—	4	1	no	—
141	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
151	SD₁₆×C₆₇	—	4	4	no	—
152	D₈×C₆₇	—	4	4	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	2	4	no	—
155	M₁₆×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	1	no	—
166	C₄⋊C₄×C₆₇	—	2	1	no	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²⋊C₄×C₆₇	—	2	4	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	1	no	—
181	C₂²⋊C₄×C₆₇	—	4	16	no	—
182	C₄²×C₆₇	—	2	4	no	—
183	C₄²×C₆₇	—	2	4	no	—
184	C₄²×C₆₇	—	2	1	no	—
185	C₄²×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	2	no	—
188	C₄²×C₆₇	—	2	2	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	4	2	no	—
191	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₆₇	—	2	1	no	—
202	Q₈×C₆₇	—	4	4	no	—
203	Q₈×C₆₇	—	4	2	no	—
204	Q₈×C₆₇	—	4	2	no	—
205	Q₈×C₆₇	—	8	2	no	—
206	Q₈×C₆₇	—	8	4	no	—
207	Q₈×C₆₇	#124	1	2	yes	complemented
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	2	1	no	—
210	D₄×C₆₇	—	4	2	no	—
211	D₄×C₆₇	—	4	2	no	—
212	D₄×C₆₇	—	4	4	no	—
213	D₄×C₆₇	—	8	2	no	—
214	D₄×C₆₇	—	8	2	no	—
215	D₄×C₆₇	—	8	2	no	—
216	D₄×C₆₇	—	8	2	no	—
217	D₄×C₆₇	#124	1	2	yes	complemented
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	4	no	—
233	C₂×C₄×C₆₇	—	8	4	no	—
234	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
235	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
236	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
237	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
241	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
242	C₈×C₆₇	—	4	2	no	—
243	C₈×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	2	1	no	—
245	C₂²×C₆₇	—	2	1	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	4	1	no	—
248	C₂²×C₆₇	—	4	1	no	—
249	C₂²×C₆₇	—	8	8	no	—
250	C₂²×C₆₇	—	8	2	no	—
251	C₂²×C₆₇	—	8	2	no	—
252	C₂²×C₆₇	—	8	2	no	—
253	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
254	C₂²×C₆₇	#548	1	2	yes	—
255	C₄×C₆₇	—	2	2	no	—
256	C₄×C₆₇	—	2	1	no	—
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	4	4	no	—
259	C₄×C₆₇	—	4	1	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	1	no	—
263	C₄×C₆₇	—	4	2	no	—
264	C₄×C₆₇	#453	1	2	yes	—
265	C₄×C₆₇	#454	1	2	yes	—
266	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
267	C₄×C₆₇	#536	1	1	yes	characteristic
268	C₄×C₆₇	#548	1	2	yes	—
269	#25924	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	2	2	no	—
271	C₂×C₆₇	—	4	2	no	—
272	C₂×C₆₇	—	16	2	no	—
273	C₂×C₆₇	#2436	1	1	yes	characteristic
274	C₂×C₆₇	C₂×#453	1	1	yes	characteristic
275	C₂×C₆₇	C₂×#454	1	1	yes	characteristic
276	#3503	C₂×C₆₇	1	1	yes	characteristic, complemented
277	#3138	C₂×C₆₇	1	1	yes	characteristic, complemented
278	#3038	C₂×C₆₇	1	1	yes	characteristic
279	#2858	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#2027	C₂×C₆₇	1	4	yes	complemented
281	#1997	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1994	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#1883	C₂×C₆₇	1	4	yes	complemented
284	C₄×#104	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₆₇	#25924	1	1	yes	characteristic, direct product factor, Sylow
286	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
287	Q₈²	—	2	1	no	—
288	#557	—	2	1	no	—
289	#554	—	2	1	no	—
290	#554	—	2	2	no	—
291	D₄×Q₈	—	2	2	no	—
292	D₄×Q₈	—	2	1	no	—
293	#544	—	2	2	no	—
294	#544	C₂²×C₆₇	1	2	yes	—
295	#533	—	2	1	no	—
296	#533	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×C₄⋊Q₈	—	2	1	no	—
298	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×#124	—	2	2	no	—
301	#520	C₂²×C₆₇	1	2	yes	—
302	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
303	#490	—	2	2	no	—
304	#486	—	2	1	no	—
305	#486	—	2	1	no	—
306	#460	—	2	1	no	—
307	#460	—	2	1	no	—
308	#459	—	2	1	no	—
309	#459	—	2	1	no	—
310	#423	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#104	C₂²×C₆₇	1	2	yes	—
312	#418	C₂²×C₆₇	1	4	yes	—
313	#418	C₂²×C₆₇	1	2	yes	—
314	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
315	#390	—	2	2	no	—
316	#390	C₂²×C₆₇	1	4	yes	—
317	C₄³	C₂²×C₆₇	1	1	yes	characteristic
318	#343	C₂²×C₆₇	1	4	yes	—
319	#339	C₂²×C₆₇	1	4	yes	—
320	#339	C₂²×C₆₇	1	2	yes	—
321	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
322	C₂²×Q₈	—	2	2	no	—
323	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
327	AGL(1,ℤ/8)	—	2	8	no	—
328	C₂×SD₁₆	—	2	2	no	—
329	C₂×D₈	—	2	2	no	—
330	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
331	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	—	2	2	no	—
334	C₄⋊Q₈	—	2	4	no	—
335	C₄⋊Q₈	—	2	2	no	—
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	2	1	no	—
341	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
342	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
343	#127	—	2	1	no	—
344	#127	—	2	1	no	—
345	#127	—	2	1	no	—
346	#127	—	2	1	no	—
347	#125	—	2	4	no	—
348	#125	—	2	2	no	—
349	#125	—	2	2	no	—
350	#125	D₄×C₆₇	1	2	yes	—
351	#124	—	4	8	no	—
352	C₂²⋊Q₈	—	2	2	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	1	no	—
357	C₂²⋊Q₈	—	2	1	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
365	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	—	2	1	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
371	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
372	#117	—	2	2	no	—
373	#117	—	2	1	no	—
374	#117	C₂³×C₆₇	1	1	yes	characteristic
375	#117	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
380	C₂×C₂²⋊C₄	—	2	8	no	—
381	C₂×C₄²	—	2	1	no	—
382	C₂×C₄²	—	2	1	no	—
383	C₂×C₄²	—	2	1	no	—
384	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
385	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	#104	—	2	4	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	4	no	—
392	#102	—	2	4	no	—
393	#97	—	2	1	no	—
394	C₄×C₈	—	2	1	no	—
395	C₂⁴	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	#41×C₆₇	1	4	yes	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	4	4	no	—
404	C₂×Q₈	—	8	4	no	—
405	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	—	4	4	no	—
413	C₂×D₄	—	4	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	—	4	1	no	—
416	C₂×D₄	—	4	1	no	—
417	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
427	SD₁₆	—	4	4	no	—
428	D₈	—	4	4	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	2	4	no	—
431	M₁₆	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
449	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
450	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	1	no	—
455	C₂²⋊C₄	—	2	1	no	—
456	C₂²⋊C₄	—	2	1	no	—
457	C₂²⋊C₄	—	4	16	no	—
458	C₄²	—	2	4	no	—
459	C₄²	—	2	4	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	1	no	—
462	C₄²	—	2	1	no	—
463	C₄²	—	2	2	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	4	2	no	—
467	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂³	—	2	1	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	4	no	—
476	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
477	Q₈	—	2	1	no	—
478	Q₈	—	4	4	no	—
479	Q₈	—	4	2	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	8	2	no	—
482	Q₈	—	8	4	no	—
483	Q₈	#124×C₆₇	1	2	yes	complemented
484	D₄	—	2	4	no	—
485	D₄	—	2	1	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	4	no	—
489	D₄	—	8	2	no	—
490	D₄	—	8	2	no	—
491	D₄	—	8	2	no	—
492	D₄	—	8	2	no	—
493	D₄	#124×C₆₇	1	2	yes	complemented
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	—	8	4	no	—
510	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
511	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
512	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
513	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
517	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
518	C₈	—	4	2	no	—
519	C₈	—	4	2	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	4	1	no	—
524	C₂²	—	4	1	no	—
525	C₂²	—	8	8	no	—
526	C₂²	—	8	2	no	—
527	C₂²	—	8	2	no	—
528	C₂²	—	8	2	no	—
529	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
530	C₂²	#548×C₆₇	1	2	yes	—
531	C₄	—	2	1	no	—
532	C₄	—	2	1	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	1	no	—
539	C₄	—	4	2	no	—
540	C₄	#453×C₆₇	1	2	yes	—
541	C₄	#454×C₆₇	1	2	yes	—
542	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
543	C₄	#536×C₆₇	1	1	yes	characteristic
544	C₄	#548×C₆₇	1	2	yes	—
545	C₂	—	2	2	no	—
546	C₂	—	4	2	no	—
547	C₂	—	16	2	no	—
548	C₂	C₆₇×#2436	1	1	yes	characteristic
549	C₂	C₂×#453×C₆₇	1	1	yes	characteristic
550	C₂	C₂×#454×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3038	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
1. C₆₇×#3503	C₂	semidirect
2. C₆₇×#3138	C₂	semidirect
4. C₆₇×#2858	C₂	semidirect
5. C₆₇×#2027	C₂	semidirect
6. C₆₇×#1997	C₂	semidirect
7. C₆₇×#1994	C₂	semidirect
8. C₆₇×#1883	C₂	semidirect
9. C₄×#104×C₆₇	C₂	semidirect
207. Q₈×C₆₇	#124	semidirect
217. D₄×C₆₇	#124	semidirect
269. #25924	C₆₇	direct (N × H)
276. #3503	C₂×C₆₇	semidirect
277. #3138	C₂×C₆₇	semidirect
279. #2858	C₂×C₆₇	semidirect
280. #2027	C₂×C₆₇	semidirect
281. #1997	C₂×C₆₇	semidirect
282. #1994	C₂×C₆₇	semidirect
283. #1883	C₂×C₆₇	semidirect
284. C₄×#104	C₂×C₆₇	semidirect
285. C₆₇	#25924	direct (N × H)
483. Q₈	#124×C₆₇	semidirect
493. D₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3503	2
C₆₇×#3138	2
C₆₇×#3038	2
C₆₇×#2858	2
C₆₇×#2027	2
C₆₇×#1997	2
C₆₇×#1994	2
C₆₇×#1883	2
C₄×#104×C₆₇	2
#25924	67

Subgroups — C67×#25924 #30705175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25924 #30705175