Subgroups — C₆₇×#19960 #30699211

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3397	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3342	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1686	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1642	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1628	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1591	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	#556×C₆₇	C₂²	1	2	yes	—
14	#550×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	1	yes	characteristic
16	#496×C₆₇	—	2	2	no	—
17	#493×C₆₇	C₂²	1	1	yes	characteristic
18	#492×C₆₇	C₂²	1	2	yes	—
19	#486×C₆₇	C₂²	1	2	yes	—
20	#485×C₆₇	—	2	2	no	—
21	#475×C₆₇	C₂²	1	2	yes	—
22	#467×C₆₇	—	2	1	no	—
23	#466×C₆₇	—	2	1	no	—
24	#463×C₆₇	—	2	2	no	—
25	#461×C₆₇	C₂²	1	2	yes	—
26	#460×C₆₇	—	2	2	no	—
27	#460×C₆₇	C₂²	1	2	yes	—
28	#459×C₆₇	C₂²	1	2	yes	—
29	#449×C₆₇	—	2	2	no	—
30	#446×C₆₇	C₂²	1	2	yes	—
31	#440×C₆₇	—	2	2	no	—
32	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	#427×C₆₇	—	2	1	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#332×C₆₇	C₂²	1	2	yes	—
40	#331×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	1	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	#127×C₆₇	—	2	2	no	—
60	#127×C₆₇	C₂³	1	2	yes	—
61	#126×C₆₇	—	4	2	no	—
62	#124×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	#121×C₆₇	—	4	2	no	—
66	C₄×Q₈×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	#117×C₆₇	—	2	2	no	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	—	2	1	no	—
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	#107×C₆₇	D₄	1	1	yes	characteristic
76	#106×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	D₄	1	2	yes	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	4	2	no	—
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	1	no	—
91	#97×C₆₇	—	2	2	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	C₂×D₄	1	2	yes	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	Q₁₆×C₆₇	—	2	2	no	—
110	SD₁₆×C₆₇	—	4	4	no	—
111	SD₁₆×C₆₇	—	4	4	no	—
112	SD₁₆×C₆₇	—	8	2	no	—
113	SD₁₆×C₆₇	—	8	2	no	—
114	D₈×C₆₇	—	2	2	no	—
115	D₈×C₆₇	—	4	2	no	—
116	D₈×C₆₇	—	8	1	no	—
117	D₈×C₆₇	—	8	2	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	4	2	no	—
123	C₂×C₈×C₆₇	—	4	2	no	—
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
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	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	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₆₇	—	2	1	no	—
136	C₂²⋊C₄×C₆₇	—	2	1	no	—
137	C₂²⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	4	2	no	—
146	Q₈×C₆₇	—	2	2	no	—
147	Q₈×C₆₇	—	2	2	no	—
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	2	1	no	—
151	D₄×C₆₇	—	2	1	no	—
152	D₄×C₆₇	—	2	1	no	—
153	D₄×C₆₇	—	2	1	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	—	16	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	8	4	no	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
174	C₈×C₆₇	—	2	1	no	—
175	C₈×C₆₇	—	2	1	no	—
176	C₈×C₆₇	—	4	1	no	—
177	C₈×C₆₇	—	4	1	no	—
178	C₈×C₆₇	—	4	1	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₈×C₆₇	#135	1	2	yes	complemented
181	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	1	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	8	2	no	—
189	C₂²×C₆₇	—	8	2	no	—
190	C₂²×C₆₇	—	8	2	no	—
191	C₂²×C₆₇	D₄²	1	1	yes	characteristic
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	4	1	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	4	2	no	—
197	C₄×C₆₇	—	8	2	no	—
198	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
199	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₆₇	#576	1	1	yes	characteristic
201	C₄×C₆₇	#577	1	1	yes	characteristic
202	#19960	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	8	1	no	—
204	C₂×C₆₇	—	8	1	no	—
205	C₂×C₆₇	—	16	2	no	—
206	C₂×C₆₇	#3272	1	1	yes	characteristic
207	C₂×C₆₇	#3287	1	1	yes	characteristic
208	C₂×C₆₇	#3290	1	1	yes	characteristic
209	#3397	C₂×C₆₇	1	2	yes	complemented
210	#3342	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#3287	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1706	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#1686	C₂×C₆₇	1	2	yes	complemented
214	#1673	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1667	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1642	C₂×C₆₇	1	2	yes	complemented
217	#1628	C₂×C₆₇	1	2	yes	complemented
218	#1591	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#19960	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#135	—	2	1	no	—
222	#556	C₂²×C₆₇	1	2	yes	—
223	#550	C₂²×C₆₇	1	2	yes	—
224	#500	C₂²×C₆₇	1	1	yes	characteristic
225	#496	—	2	2	no	—
226	#493	C₂²×C₆₇	1	1	yes	characteristic
227	#492	C₂²×C₆₇	1	2	yes	—
228	#486	C₂²×C₆₇	1	2	yes	—
229	#485	—	2	2	no	—
230	#475	C₂²×C₆₇	1	2	yes	—
231	#467	—	2	1	no	—
232	#466	—	2	1	no	—
233	#463	—	2	2	no	—
234	#461	C₂²×C₆₇	1	2	yes	—
235	#460	—	2	2	no	—
236	#460	C₂²×C₆₇	1	2	yes	—
237	#459	C₂²×C₆₇	1	2	yes	—
238	#449	—	2	2	no	—
239	#446	C₂²×C₆₇	1	2	yes	—
240	#440	—	2	2	no	—
241	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
242	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
245	#427	—	2	1	no	—
246	#335	C₂²×C₆₇	1	1	yes	characteristic
247	#335	C₂²×C₆₇	1	1	yes	characteristic
248	#332	C₂²×C₆₇	1	2	yes	—
249	#331	C₂²×C₆₇	1	2	yes	—
250	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
251	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
252	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₈²	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#41	—	2	2	no	—
255	#135	—	4	4	no	—
256	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
257	C₂×SD₁₆	—	2	2	no	—
258	C₂×SD₁₆	—	2	2	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×SD₁₆	—	2	2	no	—
261	C₂×D₈	—	2	2	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	—	2	1	no	—
264	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
268	#127	—	2	2	no	—
269	#127	C₂³×C₆₇	1	2	yes	—
270	#126	—	4	2	no	—
271	#124	—	2	2	no	—
272	C₂²⋊Q₈	—	2	2	no	—
273	#121	—	2	2	no	—
274	#121	—	4	2	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
279	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
280	#117	—	2	2	no	—
281	#107	—	2	1	no	—
282	#107	—	2	1	no	—
283	#107	D₄×C₆₇	1	1	yes	characteristic
284	#107	D₄×C₆₇	1	1	yes	characteristic
285	#106	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#103	D₄×C₆₇	1	2	yes	—
293	#102	—	2	2	no	—
294	#102	—	2	2	no	—
295	#102	—	4	2	no	—
296	#102	D₄×C₆₇	1	1	yes	characteristic
297	#102	D₄×C₆₇	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	#97	—	2	2	no	—
301	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	#41	—	4	4	no	—
305	#41	—	8	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	4	2	no	—
318	Q₁₆	—	2	2	no	—
319	SD₁₆	—	4	4	no	—
320	SD₁₆	—	4	4	no	—
321	SD₁₆	—	8	2	no	—
322	SD₁₆	—	8	2	no	—
323	D₈	—	2	2	no	—
324	D₈	—	4	2	no	—
325	D₈	—	8	1	no	—
326	D₈	—	8	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	4	2	no	—
333	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
343	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂²⋊C₄	—	2	1	no	—
345	C₂²⋊C₄	—	2	1	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	4	2	no	—
351	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	4	2	no	—
355	Q₈	—	2	2	no	—
356	Q₈	—	2	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	2	1	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	4	no	—
369	D₄	—	16	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	2	no	—
389	C₈	#135×C₆₇	1	2	yes	complemented
390	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	D₄²×C₆₇	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
408	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
409	C₄	#576×C₆₇	1	1	yes	characteristic
410	C₄	#577×C₆₇	1	1	yes	characteristic
411	C₂	—	8	1	no	—
412	C₂	—	8	1	no	—
413	C₂	—	16	2	no	—
414	C₂	C₆₇×#3272	1	1	yes	characteristic
415	C₂	C₆₇×#3287	1	1	yes	characteristic
416	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×#3397	C₂	semidirect
2. C₆₇×#3342	C₂	semidirect
3. C₆₇×#3287	C₂	semidirect
4. C₆₇×#1706	C₂	semidirect
5. C₆₇×#1686	C₂	semidirect
6. C₆₇×#1673	C₂	semidirect
7. C₆₇×#1667	C₂	semidirect
8. C₆₇×#1642	C₂	semidirect
9. C₆₇×#1628	C₂	semidirect
10. C₆₇×#1591	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
180. C₈×C₆₇	#135	semidirect
202. #19960	C₆₇	direct (N × H)
209. #3397	C₂×C₆₇	semidirect
210. #3342	C₂×C₆₇	semidirect
211. #3287	C₂×C₆₇	semidirect
212. #1706	C₂×C₆₇	semidirect
213. #1686	C₂×C₆₇	semidirect
214. #1673	C₂×C₆₇	semidirect
215. #1667	C₂×C₆₇	semidirect
216. #1642	C₂×C₆₇	semidirect
217. #1628	C₂×C₆₇	semidirect
218. #1591	C₂×C₆₇	semidirect
219. C₈×D₈	C₂×C₆₇	semidirect
220. C₆₇	#19960	direct (N × H)
389. 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×#3397	2
C₆₇×#3342	2
C₆₇×#3287	2
C₆₇×#1706	2
C₆₇×#1686	2
C₆₇×#1673	2
C₆₇×#1667	2
C₆₇×#1642	2
C₆₇×#1628	2
C₆₇×#1591	2
C₈×D₈×C₆₇	2
#19960	67

Subgroups — C67×#19960 #30699211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19960 #30699211