Subgroups — C₆₇×#19887 #30699138

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3334	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3276	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1689	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1649	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1647	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	—	2	2	no	—
13	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	—	2	1	no	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#500×C₆₇	C₂²	1	1	yes	characteristic
18	#500×C₆₇	C₂²	1	2	yes	—
19	#494×C₆₇	C₂²	1	2	yes	—
20	#480×C₆₇	—	2	2	no	—
21	#474×C₆₇	—	2	2	no	—
22	#474×C₆₇	C₂²	1	2	yes	—
23	#474×C₆₇	C₂²	1	2	yes	—
24	#470×C₆₇	—	2	2	no	—
25	#466×C₆₇	—	2	1	no	—
26	#462×C₆₇	C₂²	1	2	yes	—
27	#461×C₆₇	C₂²	1	2	yes	—
28	#452×C₆₇	—	2	2	no	—
29	#451×C₆₇	—	2	2	no	—
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
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	#436×C₆₇	C₂²	1	2	yes	—
35	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#107×C₆₇	—	2	1	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#334×C₆₇	C₂²	1	1	yes	characteristic
39	#332×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	#137×C₆₇	—	4	8	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	—	2	2	no	—
50	C₂×Q₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
53	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	#123×C₆₇	—	4	2	no	—
60	C₂²⋊Q₈×C₆₇	—	2	2	no	—
61	C₂²⋊Q₈×C₆₇	—	2	2	no	—
62	C₂²⋊Q₈×C₆₇	—	4	2	no	—
63	#121×C₆₇	—	2	2	no	—
64	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	—	2	2	no	—
74	#107×C₆₇	—	2	2	no	—
75	#107×C₆₇	—	4	2	no	—
76	#107×C₆₇	D₄	1	1	yes	characteristic
77	#106×C₆₇	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	—	2	1	no	—
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	D₄	1	1	yes	characteristic
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	D₄	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	2	yes	—
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	#41×C₆₇	—	4	4	no	—
96	#41×C₆₇	—	8	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	—	4	4	no	—
101	C₂×Q₈×C₆₇	—	8	2	no	—
102	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	Q₁₆×C₆₇	—	4	4	no	—
111	Q₁₆×C₆₇	—	4	4	no	—
112	Q₁₆×C₆₇	—	8	2	no	—
113	SD₁₆×C₆₇	—	4	4	no	—
114	SD₁₆×C₆₇	D₈	1	4	yes	complemented
115	D₈×C₆₇	—	2	2	no	—
116	M₁₆×C₆₇	—	4	4	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	4	2	no	—
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	4	2	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₆₇	—	2	2	no	—
137	C₂²⋊C₄×C₆₇	—	2	1	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	—	2	1	no	—
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	—	2	1	no	—
143	C₂³×C₆₇	—	2	2	no	—
144	Q₈×C₆₇	—	2	1	no	—
145	Q₈×C₆₇	—	4	2	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	8	4	no	—
150	Q₈×C₆₇	C₂×D₈	1	2	yes	—
151	D₄×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	1	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	8	4	no	—
155	D₄×C₆₇	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	8	4	no	—
167	C₂×C₄×C₆₇	—	8	4	no	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
171	C₈×C₆₇	—	2	2	no	—
172	C₈×C₆₇	—	2	1	no	—
173	C₈×C₆₇	—	2	1	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	1	no	—
177	C₈×C₆₇	C₂×D₈	1	2	yes	—
178	C₈×C₆₇	#137	1	2	yes	complemented
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	4	1	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	2	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	D₄²	1	1	yes	characteristic
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	1	no	—
191	C₄×C₆₇	—	8	2	no	—
192	C₄×C₆₇	—	8	2	no	—
193	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
194	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
195	C₄×C₆₇	#576	1	1	yes	characteristic
196	C₄×C₆₇	#578	1	1	yes	characteristic
197	#19887	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	4	2	no	—
199	C₂×C₆₇	—	8	2	no	—
200	C₂×C₆₇	#3276	1	1	yes	characteristic
201	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
202	C₂×C₆₇	#3293	1	1	yes	characteristic
203	Q₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#3334	C₂×C₆₇	1	1	yes	characteristic
205	#3276	C₂×C₆₇	1	2	yes	complemented
206	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1689	C₂×C₆₇	1	2	yes	complemented
208	#1674	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1673	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1649	C₂×C₆₇	1	2	yes	complemented
211	#1647	C₂×C₆₇	1	2	yes	complemented
212	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
213	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₆₇	#19887	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#137	—	2	2	no	—
216	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
217	#547	C₂²×C₆₇	1	2	yes	—
218	#500	—	2	1	no	—
219	#500	C₂²×C₆₇	1	1	yes	characteristic
220	#500	C₂²×C₆₇	1	1	yes	characteristic
221	#500	C₂²×C₆₇	1	2	yes	—
222	#494	C₂²×C₆₇	1	2	yes	—
223	#480	—	2	2	no	—
224	#474	—	2	2	no	—
225	#474	C₂²×C₆₇	1	2	yes	—
226	#474	C₂²×C₆₇	1	2	yes	—
227	#470	—	2	2	no	—
228	#466	—	2	1	no	—
229	#462	C₂²×C₆₇	1	2	yes	—
230	#461	C₂²×C₆₇	1	2	yes	—
231	#452	—	2	2	no	—
232	#451	—	2	2	no	—
233	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
235	C₄×D₈	—	2	1	no	—
236	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
237	#436	C₂²×C₆₇	1	2	yes	—
238	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×#107	—	2	1	no	—
240	#335	C₂²×C₆₇	1	1	yes	characteristic
241	#334	C₂²×C₆₇	1	1	yes	characteristic
242	#332	C₂²×C₆₇	1	2	yes	—
243	#332	C₂²×C₆₇	1	2	yes	—
244	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
245	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
247	C₈²	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#41	—	2	2	no	—
249	C₂²×Q₈	—	2	2	no	—
250	#137	—	4	8	no	—
251	C₂×Q₁₆	—	2	2	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
256	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
257	C₂×M₁₆	—	2	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	2	no	—
260	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
261	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
262	#123	—	4	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	2	2	no	—
265	C₂²⋊Q₈	—	4	2	no	—
266	#121	—	2	2	no	—
267	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
270	C₄×D₄	C₂³×C₆₇	1	2	yes	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	2	no	—
277	#107	—	2	2	no	—
278	#107	—	4	2	no	—
279	#107	D₄×C₆₇	1	1	yes	characteristic
280	#106	D₄×C₆₇	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
284	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
285	#103	—	2	2	no	—
286	#103	—	2	2	no	—
287	#103	D₄×C₆₇	1	1	yes	characteristic
288	#102	—	2	2	no	—
289	#102	—	2	2	no	—
290	#102	D₄×C₆₇	1	2	yes	—
291	#102	D₄×C₆₇	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	2	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	#41	—	4	4	no	—
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	—	8	2	no	—
305	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
308	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	Q₁₆	—	4	4	no	—
314	Q₁₆	—	4	4	no	—
315	Q₁₆	—	8	2	no	—
316	SD₁₆	—	4	4	no	—
317	SD₁₆	D₈×C₆₇	1	4	yes	complemented
318	D₈	—	2	2	no	—
319	M₁₆	—	4	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	2	no	—
325	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	Q₈	—	8	4	no	—
353	Q₈	C₂×D₈×C₆₇	1	2	yes	—
354	D₄	—	2	2	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	2	no	—
357	D₄	—	8	4	no	—
358	D₄	C₂×D₈×C₆₇	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	—	8	4	no	—
371	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	1	no	—
379	C₈	—	4	1	no	—
380	C₈	C₂×D₈×C₆₇	1	2	yes	—
381	C₈	#137×C₆₇	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	D₄²×C₆₇	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
397	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
398	C₄	#576×C₆₇	1	1	yes	characteristic
399	C₄	#578×C₆₇	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	8	2	no	—
402	C₂	C₆₇×#3276	1	1	yes	characteristic
403	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
404	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3334	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. Q₈×D₈×C₆₇	C₂	semidirect
3. C₆₇×#3276	C₂	semidirect
4. C₆₇×#1713	C₂	semidirect
5. C₆₇×#1689	C₂	semidirect
6. C₆₇×#1674	C₂	semidirect
7. C₆₇×#1673	C₂	semidirect
8. C₆₇×#1649	C₂	semidirect
9. C₆₇×#1647	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×SD₁₆×C₆₇	C₂	semidirect
114. SD₁₆×C₆₇	D₈	semidirect
178. C₈×C₆₇	#137	semidirect
197. #19887	C₆₇	direct (N × H)
203. Q₈×D₈	C₂×C₆₇	semidirect
205. #3276	C₂×C₆₇	semidirect
206. #1713	C₂×C₆₇	semidirect
207. #1689	C₂×C₆₇	semidirect
208. #1674	C₂×C₆₇	semidirect
209. #1673	C₂×C₆₇	semidirect
210. #1649	C₂×C₆₇	semidirect
211. #1647	C₂×C₆₇	semidirect
212. #1589	C₂×C₆₇	semidirect
213. C₈×SD₁₆	C₂×C₆₇	semidirect
214. C₆₇	#19887	direct (N × H)
317. SD₁₆	D₈×C₆₇	semidirect
381. C₈	#137×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₈×C₆₇	2
C₆₇×#3334	2
C₆₇×#3276	2
C₆₇×#1713	2
C₆₇×#1689	2
C₆₇×#1674	2
C₆₇×#1673	2
C₆₇×#1649	2
C₆₇×#1647	2
C₆₇×#1589	2
C₈×SD₁₆×C₆₇	2
#19887	67

Subgroups — C67×#19887 #30699138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19887 #30699138