Subgroups — C₆₇×#10055 #30689306

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#331×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#124×C₆₇	—	2	1	no	—
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#492×C₆₇	—	2	2	no	—
14	#492×C₆₇	—	2	2	no	—
15	#477×C₆₇	—	2	2	no	—
16	#477×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	—	2	1	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₆₇	—	2	1	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#394×C₆₇	—	2	2	no	—
23	#390×C₆₇	—	2	2	no	—
24	#390×C₆₇	—	2	1	no	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	#340×C₆₇	—	2	2	no	—
27	#331×C₆₇	—	2	2	no	—
28	Q₈⋊C₈×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	D₄	1	2	yes	—
34	C₂×SD₁₆×C₆₇	—	2	8	no	—
35	C₂×SD₁₆×C₆₇	—	4	8	no	—
36	C₂²×C₈×C₆₇	—	2	1	no	—
37	C₂²×C₈×C₆₇	—	2	1	no	—
38	C₂²×C₈×C₆₇	—	2	1	no	—
39	C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₄⋊Q₈×C₆₇	—	2	1	no	—
41	C₄⋊Q₈×C₆₇	—	2	1	no	—
42	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
43	#127×C₆₇	—	2	1	no	—
44	#127×C₆₇	—	2	1	no	—
45	#127×C₆₇	—	2	1	no	—
46	#127×C₆₇	—	2	1	no	—
47	#124×C₆₇	—	4	2	no	—
48	#124×C₆₇	—	4	2	no	—
49	C₄×Q₈×C₆₇	—	2	1	no	—
50	C₄×Q₈×C₆₇	—	2	1	no	—
51	C₄×Q₈×C₆₇	—	2	1	no	—
52	C₄×Q₈×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₄²×C₆₇	—	2	1	no	—
61	C₂×C₄²×C₆₇	—	2	1	no	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	#107×C₆₇	—	2	4	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
68	#103×C₆₇	—	2	4	no	—
69	C₄×C₈×C₆₇	—	2	1	no	—
70	C₄×C₈×C₆₇	D₄	1	2	yes	—
71	#95×C₆₇	—	2	1	no	—
72	C₂⁴×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	1	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	2	1	no	—
77	C₂×Q₈×C₆₇	—	2	1	no	—
78	C₂×Q₈×C₆₇	—	2	1	no	—
79	C₂×Q₈×C₆₇	—	4	4	no	—
80	C₂×Q₈×C₆₇	—	4	1	no	—
81	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	complemented
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	4	4	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	4	1	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	SD₁₆×C₆₇	—	4	16	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄⋊C₄×C₆₇	—	4	1	no	—
119	C₄⋊C₄×C₆₇	—	4	1	no	—
120	C₂²⋊C₄×C₆₇	—	4	8	no	—
121	C₂²⋊C₄×C₆₇	—	8	4	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₆₇	#41	1	1	yes	characteristic
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	4	no	—
135	C₂³×C₆₇	#120	1	1	yes	characteristic
136	Q₈×C₆₇	—	2	2	no	—
137	Q₈×C₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	4	4	no	—
139	Q₈×C₆₇	—	4	1	no	—
140	Q₈×C₆₇	—	4	1	no	—
141	Q₈×C₆₇	—	8	2	no	—
142	D₄×C₆₇	—	4	4	no	—
143	D₄×C₆₇	—	8	2	no	—
144	D₄×C₆₇	—	8	1	no	—
145	D₄×C₆₇	—	8	1	no	—
146	D₄×C₆₇	—	8	2	no	—
147	D₄×C₆₇	—	8	1	no	—
148	D₄×C₆₇	—	8	1	no	—
149	D₄×C₆₇	—	8	1	no	—
150	D₄×C₆₇	—	8	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	8	2	no	—
162	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
165	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
167	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
170	C₂×C₄×C₆₇	#135	1	2	yes	—
171	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#137	1	2	yes	—
174	C₈×C₆₇	—	2	4	no	—
175	C₈×C₆₇	—	4	2	no	—
176	C₈×C₆₇	—	8	1	no	—
177	C₂²×C₆₇	—	8	4	no	—
178	C₂²×C₆₇	—	8	4	no	—
179	C₂²×C₆₇	—	8	2	no	—
180	C₂²×C₆₇	—	8	2	no	—
181	C₂²×C₆₇	—	8	2	no	—
182	C₂²×C₆₇	#392	1	1	yes	characteristic
183	C₂²×C₆₇	#450	1	1	yes	characteristic
184	C₂²×C₆₇	#452	1	1	yes	characteristic
185	C₂²×C₆₇	#452	1	2	yes	—
186	C₂²×C₆₇	#453	1	2	yes	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	—	8	1	no	—
192	C₄×C₆₇	—	8	2	no	—
193	C₄×C₆₇	#448	1	1	yes	characteristic
194	C₄×C₆₇	#449	1	1	yes	characteristic
195	C₄×C₆₇	#464	1	2	yes	—
196	C₄×C₆₇	#465	1	1	yes	characteristic
197	C₄×C₆₇	#468	1	1	yes	characteristic
198	C₄×C₆₇	#495	1	1	yes	characteristic
199	C₄×C₆₇	#497	1	1	yes	characteristic
200	#10055	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	16	2	no	—
202	C₂×C₆₇	#1630	1	2	yes	—
203	C₂×C₆₇	#1631	1	2	yes	—
204	C₂×C₆₇	#2000	1	1	yes	characteristic
205	C₂×C₆₇	#2002	1	1	yes	characteristic
206	C₂×C₆₇	#2015	1	1	yes	characteristic
207	C₂×#492	C₂×C₆₇	1	1	yes	characteristic
208	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#2548	C₂×C₆₇	1	1	yes	characteristic
210	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
211	C₂×#331	C₂×C₆₇	1	1	yes	characteristic
212	C₂×Q₈⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#1279	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₆₇	#10055	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×SD₁₆	—	2	2	no	—
216	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#124	—	2	1	no	—
219	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
220	#492	—	2	2	no	—
221	#492	—	2	2	no	—
222	#477	—	2	2	no	—
223	#477	—	2	2	no	—
224	C₂×#107	—	2	1	no	—
225	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#103	—	2	1	no	—
228	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
229	#394	—	2	2	no	—
230	#390	—	2	2	no	—
231	#390	—	2	1	no	—
232	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
233	#340	—	2	2	no	—
234	#331	—	2	2	no	—
235	Q₈⋊C₈	—	2	2	no	—
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
238	C₂²×D₄	—	2	1	no	—
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	D₄×C₆₇	1	2	yes	—
241	C₂×SD₁₆	—	2	8	no	—
242	C₂×SD₁₆	—	4	8	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
250	#127	—	2	1	no	—
251	#127	—	2	1	no	—
252	#127	—	2	1	no	—
253	#127	—	2	1	no	—
254	#124	—	4	2	no	—
255	#124	—	4	2	no	—
256	C₄×Q₈	—	2	1	no	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
265	C₂×C₂²⋊C₄	—	2	4	no	—
266	C₂×C₂²⋊C₄	—	2	4	no	—
267	C₂×C₄²	—	2	1	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
270	#107	—	2	4	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
275	#103	—	2	4	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	D₄×C₆₇	1	2	yes	—
278	#95	—	2	1	no	—
279	C₂⁴	—	2	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	4	4	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	SD₁₆×C₆₇	1	2	yes	complemented
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	4	4	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	SD₁₆	—	4	16	no	—
310	C₂×C₈	—	2	4	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₂²⋊C₄	—	4	8	no	—
328	C₂²⋊C₄	—	8	4	no	—
329	C₄²	—	4	1	no	—
330	C₄²	—	4	1	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₄²	#41×C₆₇	1	1	yes	characteristic
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	4	no	—
340	C₂³	—	4	4	no	—
341	C₂³	—	4	2	no	—
342	C₂³	#120×C₆₇	1	1	yes	characteristic
343	Q₈	—	2	2	no	—
344	Q₈	—	2	2	no	—
345	Q₈	—	4	4	no	—
346	Q₈	—	4	1	no	—
347	Q₈	—	4	1	no	—
348	Q₈	—	8	2	no	—
349	D₄	—	4	4	no	—
350	D₄	—	8	2	no	—
351	D₄	—	8	1	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	1	no	—
355	D₄	—	8	1	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
371	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
372	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
373	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
374	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
376	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
377	C₂×C₄	#135×C₆₇	1	2	yes	—
378	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
379	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
380	C₂×C₄	#137×C₆₇	1	2	yes	—
381	C₈	—	2	4	no	—
382	C₈	—	4	2	no	—
383	C₈	—	8	1	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	4	no	—
388	C₂²	—	8	4	no	—
389	C₂²	#392×C₆₇	1	1	yes	characteristic
390	C₂²	#450×C₆₇	1	1	yes	characteristic
391	C₂²	#452×C₆₇	1	1	yes	characteristic
392	C₂²	#452×C₆₇	1	2	yes	—
393	C₂²	#453×C₆₇	1	2	yes	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	4	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	#448×C₆₇	1	1	yes	characteristic
401	C₄	#449×C₆₇	1	1	yes	characteristic
402	C₄	#464×C₆₇	1	2	yes	—
403	C₄	#465×C₆₇	1	1	yes	characteristic
404	C₄	#468×C₆₇	1	1	yes	characteristic
405	C₄	#495×C₆₇	1	1	yes	characteristic
406	C₄	#497×C₆₇	1	1	yes	characteristic
407	C₂	—	16	2	no	—
408	C₂	C₆₇×#1630	1	2	yes	—
409	C₂	C₆₇×#1631	1	2	yes	—
410	C₂	C₆₇×#2000	1	1	yes	characteristic
411	C₂	C₆₇×#2002	1	1	yes	characteristic
412	C₂	C₆₇×#2015	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#477×C₆₇	C₂	semidirect
4. C₆₇×#1987	C₂	semidirect
6. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
7. C₆₇×#1279	C₂	semidirect
81. C₂×Q₈×C₆₇	SD₁₆	semidirect
200. #10055	C₆₇	direct (N × H)
208. C₂×#477	C₂×C₆₇	semidirect
210. #1987	C₂×C₆₇	semidirect
212. C₂×Q₈⋊C₈	C₂×C₆₇	semidirect
213. #1279	C₂×C₆₇	semidirect
214. C₆₇	#10055	direct (N × H)
288. C₂×Q₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₆₇	2
C₂×#477×C₆₇	2
C₆₇×#2548	2
C₆₇×#1987	2
C₂×#331×C₆₇	2
C₂×Q₈⋊C₈×C₆₇	2
C₆₇×#1279	2
#10055	67

Subgroups — C67×#10055 #30689306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10055 #30689306