Subgroups — C₆₇×#19976 #30699227

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3295	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1677	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1629	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1622	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	—	2	1	no	—
13	#554×C₆₇	C₂²	1	2	yes	complemented
14	D₄²×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#499×C₆₇	—	2	1	no	—
17	#493×C₆₇	C₂²	1	1	yes	characteristic
18	#492×C₆₇	C₂²	1	2	yes	complemented
19	#486×C₆₇	C₂²	1	2	yes	complemented
20	#481×C₆₇	—	2	2	no	—
21	#478×C₆₇	—	2	2	no	—
22	#476×C₆₇	C₂²	1	2	yes	—
23	#476×C₆₇	C₂²	1	2	yes	complemented
24	#466×C₆₇	—	2	1	no	—
25	#465×C₆₇	—	2	2	no	—
26	#460×C₆₇	C₂²	1	2	yes	complemented
27	#459×C₆₇	C₂²	1	2	yes	—
28	#450×C₆₇	—	2	2	no	—
29	#447×C₆₇	—	2	2	no	—
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₆₇	—	2	1	no	—
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#102×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#334×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#333×C₆₇	C₂²	1	2	yes	complemented
40	#331×C₆₇	C₂²	1	2	yes	complemented
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	—	4	2	no	—
49	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
50	C₂×D₈×C₆₇	—	2	2	no	—
51	C₂×D₈×C₆₇	—	2	1	no	—
52	C₂×D₈×C₆₇	—	2	1	no	—
53	C₂×D₈×C₆₇	—	4	4	no	—
54	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	#127×C₆₇	C₂³	1	2	yes	—
59	#125×C₆₇	—	2	2	no	—
60	C₂²⋊Q₈×C₆₇	—	2	2	no	—
61	C₂²⋊Q₈×C₆₇	—	4	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	#121×C₆₇	—	2	2	no	—
64	#120×C₆₇	—	4	2	no	—
65	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₆₇	—	2	2	no	—
67	C₄×D₄×C₆₇	C₂³	1	2	yes	—
68	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	D₄	1	1	yes	characteristic
73	#106×C₆₇	—	2	2	no	—
74	#106×C₆₇	—	2	2	no	—
75	#106×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	1	no	—
77	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#103×C₆₇	D₄	1	1	yes	characteristic
81	#102×C₆₇	—	2	2	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	2	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₄×C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂⁴×C₆₇	—	4	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂×D₄×C₆₇	—	8	2	no	—
107	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
112	SD₁₆×C₆₇	—	2	2	no	—
113	SD₁₆×C₆₇	—	8	4	no	—
114	SD₁₆×C₆₇	—	8	2	no	—
115	D₈×C₆₇	—	4	4	no	—
116	D₈×C₆₇	—	4	2	no	—
117	D₈×C₆₇	—	8	1	no	—
118	D₈×C₆₇	SD₁₆	1	4	yes	complemented
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	4	2	no	—
124	C₂×C₈×C₆₇	—	8	2	no	—
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₆₇	—	2	1	no	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₄²×C₆₇	—	2	1	no	—
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	C₂³×C₆₇	—	8	4	no	—
147	C₂³×C₆₇	—	8	4	no	—
148	C₂³×C₆₇	—	16	2	no	—
149	Q₈×C₆₇	—	2	1	no	—
150	Q₈×C₆₇	—	2	1	no	—
151	Q₈×C₆₇	—	8	2	no	—
152	D₄×C₆₇	—	2	1	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	4	2	no	—
157	D₄×C₆₇	—	4	2	no	—
158	D₄×C₆₇	—	8	2	no	—
159	D₄×C₆₇	—	8	4	no	—
160	D₄×C₆₇	—	8	4	no	—
161	D₄×C₆₇	C₂×SD₁₆	1	4	yes	—
162	D₄×C₆₇	#135	1	2	yes	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	16	2	no	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	—	2	1	no	—
179	C₈×C₆₇	—	2	1	no	—
180	C₈×C₆₇	—	4	1	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₈×C₆₇	—	4	1	no	—
183	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	—	8	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	16	8	no	—
194	C₂²×C₆₇	D₄²	1	1	yes	characteristic
195	C₄×C₆₇	—	2	1	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	—	4	1	no	—
198	C₄×C₆₇	—	4	1	no	—
199	C₄×C₆₇	—	4	1	no	—
200	C₄×C₆₇	—	8	2	no	—
201	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
202	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
203	C₄×C₆₇	#576	1	1	yes	characteristic
204	C₄×C₆₇	#577	1	1	yes	characteristic
205	#19976	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₆₇	—	4	4	no	—
207	C₂×C₆₇	—	4	2	no	—
208	C₂×C₆₇	—	16	2	no	—
209	C₂×C₆₇	#3280	1	1	yes	characteristic
210	C₂×C₆₇	D₄×SD₁₆	1	1	yes	characteristic
211	C₂×C₆₇	#3292	1	1	yes	characteristic
212	#3380	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#3295	C₂×C₆₇	1	2	yes	complemented
214	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic
215	#1706	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1677	C₂×C₆₇	1	2	yes	complemented
217	#1675	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1668	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#1629	C₂×C₆₇	1	2	yes	complemented
220	#1622	C₂×C₆₇	1	2	yes	complemented
221	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
222	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₆₇	#19976	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²×D₈	—	2	1	no	—
225	#554	C₂²×C₆₇	1	2	yes	complemented
226	D₄²	C₂²×C₆₇	1	2	yes	—
227	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
228	#499	—	2	1	no	—
229	#493	C₂²×C₆₇	1	1	yes	characteristic
230	#492	C₂²×C₆₇	1	2	yes	complemented
231	#486	C₂²×C₆₇	1	2	yes	complemented
232	#481	—	2	2	no	—
233	#478	—	2	2	no	—
234	#476	C₂²×C₆₇	1	2	yes	—
235	#476	C₂²×C₆₇	1	2	yes	complemented
236	#466	—	2	1	no	—
237	#465	—	2	2	no	—
238	#460	C₂²×C₆₇	1	2	yes	complemented
239	#459	C₂²×C₆₇	1	2	yes	—
240	#450	—	2	2	no	—
241	#447	—	2	2	no	—
242	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
243	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
244	C₄×D₈	—	2	1	no	—
245	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
247	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
248	C₂×#102	—	2	2	no	—
249	#335	C₂²×C₆₇	1	1	yes	characteristic, complemented
250	#334	C₂²×C₆₇	1	1	yes	characteristic, complemented
251	#333	C₂²×C₆₇	1	2	yes	complemented
252	#331	C₂²×C₆₇	1	2	yes	complemented
253	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
254	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
255	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
256	C₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	—	2	2	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×SD₁₆	—	4	2	no	—
261	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
262	C₂×D₈	—	2	2	no	—
263	C₂×D₈	—	2	1	no	—
264	C₂×D₈	—	2	1	no	—
265	C₂×D₈	—	4	4	no	—
266	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
270	#127	C₂³×C₆₇	1	2	yes	—
271	#125	—	2	2	no	—
272	C₂²⋊Q₈	—	2	2	no	—
273	C₂²⋊Q₈	—	4	2	no	—
274	#121	—	2	2	no	—
275	#121	—	2	2	no	—
276	#120	—	4	2	no	—
277	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
280	C₄×D₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	#107	—	2	1	no	—
283	#107	—	2	1	no	—
284	#107	D₄×C₆₇	1	1	yes	characteristic
285	#106	—	2	2	no	—
286	#106	—	2	2	no	—
287	#106	D₄×C₆₇	1	1	yes	characteristic
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
291	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
292	#103	D₄×C₆₇	1	1	yes	characteristic
293	#102	—	2	2	no	—
294	#102	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	D₄×C₆₇	1	1	yes	characteristic
300	#102	D₄×C₆₇	1	2	yes	—
301	C₂²⋊C₈	—	2	1	no	—
302	C₂²⋊C₈	—	2	2	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂⁴	—	4	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	4	no	—
316	C₂×D₄	—	4	4	no	—
317	C₂×D₄	—	8	2	no	—
318	C₂×D₄	—	8	2	no	—
319	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
320	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	4	2	no	—
324	SD₁₆	—	2	2	no	—
325	SD₁₆	—	8	4	no	—
326	SD₁₆	—	8	2	no	—
327	D₈	—	4	4	no	—
328	D₈	—	4	2	no	—
329	D₈	—	8	1	no	—
330	D₈	SD₁₆×C₆₇	1	4	yes	complemented
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	—	8	2	no	—
337	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
347	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
348	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₄²	—	2	1	no	—
354	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
355	C₂³	—	2	2	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	8	4	no	—
359	C₂³	—	8	4	no	—
360	C₂³	—	16	2	no	—
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	1	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	8	2	no	—
371	D₄	—	8	4	no	—
372	D₄	—	8	4	no	—
373	D₄	C₂×SD₁₆×C₆₇	1	4	yes	—
374	D₄	#135×C₆₇	1	2	yes	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	16	2	no	—
385	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
387	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
388	C₈	—	2	2	no	—
389	C₈	—	2	1	no	—
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	1	no	—
395	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	16	8	no	—
406	C₂²	D₄²×C₆₇	1	1	yes	characteristic
407	C₄	—	2	1	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	8	2	no	—
413	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
414	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
415	C₄	#576×C₆₇	1	1	yes	characteristic
416	C₄	#577×C₆₇	1	1	yes	characteristic
417	C₂	—	4	2	no	—
418	C₂	—	4	4	no	—
419	C₂	—	16	2	no	—
420	C₂	C₆₇×#3280	1	1	yes	characteristic
421	C₂	D₄×SD₁₆×C₆₇	1	1	yes	characteristic
422	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	D₄×D₈×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3380	C₂	semidirect
2. C₆₇×#3295	C₂	semidirect
4. C₆₇×#1706	C₂	semidirect
5. C₆₇×#1677	C₂	semidirect
6. C₆₇×#1675	C₂	semidirect
7. C₆₇×#1668	C₂	semidirect
8. C₆₇×#1629	C₂	semidirect
9. C₆₇×#1622	C₂	semidirect
10. C₈×SD₁₆×C₆₇	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
13. #554×C₆₇	C₂²	semidirect
15. #500×C₆₇	C₂²	semidirect
18. #492×C₆₇	C₂²	semidirect
19. #486×C₆₇	C₂²	semidirect
23. #476×C₆₇	C₂²	semidirect
26. #460×C₆₇	C₂²	semidirect
30. C₈×Q₈×C₆₇	C₂²	semidirect
35. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #334×C₆₇	C₂²	semidirect
39. #333×C₆₇	C₂²	semidirect
40. #331×C₆₇	C₂²	semidirect
41. Q₈⋊C₈×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. C₈²×C₆₇	C₂²	semidirect
118. D₈×C₆₇	SD₁₆	semidirect
205. #19976	C₆₇	direct (N × H)
212. #3380	C₂×C₆₇	semidirect
213. #3295	C₂×C₆₇	semidirect
215. #1706	C₂×C₆₇	semidirect
216. #1677	C₂×C₆₇	semidirect
217. #1675	C₂×C₆₇	semidirect
218. #1668	C₂×C₆₇	semidirect
219. #1629	C₂×C₆₇	semidirect
220. #1622	C₂×C₆₇	semidirect
221. C₈×SD₁₆	C₂×C₆₇	semidirect
222. C₈×D₈	C₂×C₆₇	semidirect
223. C₆₇	#19976	direct (N × H)
225. #554	C₂²×C₆₇	semidirect
227. #500	C₂²×C₆₇	semidirect
230. #492	C₂²×C₆₇	semidirect
231. #486	C₂²×C₆₇	semidirect
235. #476	C₂²×C₆₇	semidirect
238. #460	C₂²×C₆₇	semidirect
242. C₈×Q₈	C₂²×C₆₇	semidirect
247. C₈×D₄	C₂²×C₆₇	semidirect
249. #335	C₂²×C₆₇	semidirect
250. #334	C₂²×C₆₇	semidirect
251. #333	C₂²×C₆₇	semidirect
252. #331	C₂²×C₆₇	semidirect
253. Q₈⋊C₈	C₂²×C₆₇	semidirect
255. D₄⋊C₈	C₂²×C₆₇	semidirect
256. C₈²	C₂²×C₆₇	semidirect
330. D₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3380	2
C₆₇×#3295	2
D₄×D₈×C₆₇	2
C₆₇×#1706	2
C₆₇×#1677	2
C₆₇×#1675	2
C₆₇×#1668	2
C₆₇×#1629	2
C₆₇×#1622	2
C₈×SD₁₆×C₆₇	2
C₈×D₈×C₆₇	2
#19976	67

Subgroups — C67×#19976 #30699227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19976 #30699227