Subgroups — C₆₇×#19953 #30699204

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3354	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3341	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1691	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1646	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1635	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	#555×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	1	yes	characteristic
16	#498×C₆₇	C₂²	1	2	yes	—
17	#494×C₆₇	C₂²	1	1	yes	characteristic
18	#487×C₆₇	C₂²	1	2	yes	—
19	#482×C₆₇	—	2	2	no	—
20	#482×C₆₇	—	2	2	no	—
21	#481×C₆₇	—	2	2	no	—
22	#476×C₆₇	C₂²	1	2	yes	—
23	#476×C₆₇	C₂²	1	2	yes	—
24	#474×C₆₇	C₂²	1	2	yes	—
25	#469×C₆₇	—	2	2	no	—
26	#467×C₆₇	—	2	1	no	—
27	#466×C₆₇	—	2	1	no	—
28	#461×C₆₇	C₂²	1	2	yes	—
29	#452×C₆₇	—	2	2	no	—
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
32	#436×C₆₇	C₂²	1	2	yes	—
33	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	#428×C₆₇	—	2	2	no	—
36	#427×C₆₇	—	2	1	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#333×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	D₄⋊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	#135×C₆₇	—	4	4	no	—
47	C₂×Q₁₆×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
57	#124×C₆₇	—	4	2	no	—
58	#123×C₆₇	—	4	2	no	—
59	C₂²⋊Q₈×C₆₇	—	2	2	no	—
60	#121×C₆₇	—	2	2	no	—
61	#121×C₆₇	—	2	2	no	—
62	#121×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	C₂³	1	2	yes	—
64	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	#117×C₆₇	—	2	2	no	—
67	#117×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	2	no	—
72	#107×C₆₇	D₄	1	1	yes	characteristic
73	#107×C₆₇	D₄	1	1	yes	characteristic
74	#106×C₆₇	—	2	2	no	—
75	#106×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	#106×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	4	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	D₄	1	2	yes	—
87	#102×C₆₇	D₄	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊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×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	Q₁₆×C₆₇	—	4	2	no	—
109	Q₁₆×C₆₇	—	8	1	no	—
110	SD₁₆×C₆₇	—	4	4	no	—
111	D₈×C₆₇	—	2	2	no	—
112	D₈×C₆₇	—	2	2	no	—
113	M₁₆×C₆₇	—	4	4	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₆₇	—	2	1	no	—
133	C₂²⋊C₄×C₆₇	—	2	2	no	—
134	C₂²⋊C₄×C₆₇	—	2	1	no	—
135	C₂²⋊C₄×C₆₇	—	4	2	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₄²×C₆₇	—	4	2	no	—
139	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂³×C₆₇	—	2	1	no	—
141	C₂³×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	Q₈×C₆₇	—	4	2	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	Q₈×C₆₇	—	8	2	no	—
146	D₄×C₆₇	—	2	1	no	—
147	D₄×C₆₇	—	2	1	no	—
148	D₄×C₆₇	—	2	2	no	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	2	1	no	—
151	D₄×C₆₇	—	2	1	no	—
152	D₄×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	16	2	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	8	4	no	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₆₇	—	2	1	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	4	1	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₈×C₆₇	#135	1	2	yes	complemented
176	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
177	C₂²×C₆₇	—	4	1	no	—
178	C₂²×C₆₇	—	4	1	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	D₄²	1	1	yes	characteristic
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	4	1	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	8	2	no	—
192	C₄×C₆₇	—	8	2	no	—
193	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
194	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄×C₆₇	#577	1	1	yes	characteristic
196	C₄×C₆₇	#578	1	1	yes	characteristic
197	#19953	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₆₇	—	8	1	no	—
199	C₂×C₆₇	—	8	1	no	—
200	C₂×C₆₇	—	8	2	no	—
201	C₂×C₆₇	#3272	1	1	yes	characteristic
202	C₂×C₆₇	#3283	1	1	yes	characteristic
203	C₂×C₆₇	#3290	1	1	yes	characteristic
204	#3354	C₂×C₆₇	1	2	yes	complemented
205	#3341	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#3283	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1707	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1691	C₂×C₆₇	1	2	yes	complemented
209	#1675	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#1646	C₂×C₆₇	1	2	yes	complemented
212	#1635	C₂×C₆₇	1	2	yes	complemented
213	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₆₇	#19953	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×#135	—	2	1	no	—
217	#555	C₂²×C₆₇	1	2	yes	—
218	#547	C₂²×C₆₇	1	2	yes	—
219	#500	C₂²×C₆₇	1	1	yes	characteristic
220	#498	C₂²×C₆₇	1	2	yes	—
221	#494	C₂²×C₆₇	1	1	yes	characteristic
222	#487	C₂²×C₆₇	1	2	yes	—
223	#482	—	2	2	no	—
224	#482	—	2	2	no	—
225	#481	—	2	2	no	—
226	#476	C₂²×C₆₇	1	2	yes	—
227	#476	C₂²×C₆₇	1	2	yes	—
228	#474	C₂²×C₆₇	1	2	yes	—
229	#469	—	2	2	no	—
230	#467	—	2	1	no	—
231	#466	—	2	1	no	—
232	#461	C₂²×C₆₇	1	2	yes	—
233	#452	—	2	2	no	—
234	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
236	#436	C₂²×C₆₇	1	2	yes	—
237	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
238	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
239	#428	—	2	2	no	—
240	#427	—	2	1	no	—
241	#335	C₂²×C₆₇	1	1	yes	characteristic
242	#335	C₂²×C₆₇	1	1	yes	characteristic
243	#333	C₂²×C₆₇	1	2	yes	—
244	#332	C₂²×C₆₇	1	2	yes	—
245	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
247	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
248	C₈²	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#41	—	2	2	no	—
250	#135	—	4	4	no	—
251	C₂×Q₁₆	—	2	1	no	—
252	C₂×Q₁₆	—	2	1	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
255	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
256	C₂×M₁₆	—	2	2	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
260	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
261	#124	—	4	2	no	—
262	#123	—	4	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	#121	—	2	2	no	—
265	#121	—	2	2	no	—
266	#121	—	2	2	no	—
267	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₄×D₄	C₂³×C₆₇	1	2	yes	—
269	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
270	#117	—	2	2	no	—
271	#117	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	#107	—	2	1	no	—
274	#107	—	2	1	no	—
275	#107	—	2	2	no	—
276	#107	D₄×C₆₇	1	1	yes	characteristic
277	#107	D₄×C₆₇	1	1	yes	characteristic
278	#106	—	2	2	no	—
279	#106	—	2	2	no	—
280	#106	—	2	2	no	—
281	#106	—	2	2	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	—	4	2	no	—
287	#102	—	2	2	no	—
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	#102	D₄×C₆₇	1	1	yes	characteristic
293	C₂²⋊C₈	—	2	1	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	2	no	—
296	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	#41	—	4	4	no	—
300	#41	—	8	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	4	2	no	—
312	Q₁₆	—	4	2	no	—
313	Q₁₆	—	8	1	no	—
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	M₁₆	—	4	4	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
334	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂²⋊C₄	—	2	1	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	1	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₄²	—	4	2	no	—
343	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	8	2	no	—
350	D₄	—	2	1	no	—
351	D₄	—	2	1	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	8	4	no	—
357	D₄	—	16	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
373	C₈	—	2	1	no	—
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	#135×C₆₇	1	2	yes	complemented
380	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	1	no	—
390	C₂²	D₄²×C₆₇	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	8	2	no	—
397	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
398	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
399	C₄	#577×C₆₇	1	1	yes	characteristic
400	C₄	#578×C₆₇	1	1	yes	characteristic
401	C₂	—	8	1	no	—
402	C₂	—	8	1	no	—
403	C₂	—	8	2	no	—
404	C₂	C₆₇×#3272	1	1	yes	characteristic
405	C₂	C₆₇×#3283	1	1	yes	characteristic
406	C₂	C₆₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
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×#3354	C₂	semidirect
2. C₆₇×#3341	C₂	semidirect
3. C₆₇×#3283	C₂	semidirect
4. C₆₇×#1707	C₂	semidirect
5. C₆₇×#1691	C₂	semidirect
6. C₆₇×#1675	C₂	semidirect
7. C₆₇×#1669	C₂	semidirect
8. C₆₇×#1646	C₂	semidirect
9. C₆₇×#1635	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
175. C₈×C₆₇	#135	semidirect
197. #19953	C₆₇	direct (N × H)
204. #3354	C₂×C₆₇	semidirect
205. #3341	C₂×C₆₇	semidirect
206. #3283	C₂×C₆₇	semidirect
207. #1707	C₂×C₆₇	semidirect
208. #1691	C₂×C₆₇	semidirect
209. #1675	C₂×C₆₇	semidirect
210. #1669	C₂×C₆₇	semidirect
211. #1646	C₂×C₆₇	semidirect
212. #1635	C₂×C₆₇	semidirect
213. #1589	C₂×C₆₇	semidirect
214. C₈×D₈	C₂×C₆₇	semidirect
215. C₆₇	#19953	direct (N × H)
379. C₈	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3354	2
C₆₇×#3341	2
C₆₇×#3283	2
C₆₇×#1707	2
C₆₇×#1691	2
C₆₇×#1675	2
C₆₇×#1669	2
C₆₇×#1646	2
C₆₇×#1635	2
C₆₇×#1589	2
C₈×D₈×C₆₇	2
#19953	67

Subgroups — C67×#19953 #30699204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19953 #30699204