Subgroups — C₆₇×#19398 #30698649

Order 17152.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3205	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2982	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1682	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1668	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₇	—	2	1	no	—
11	C₂²×SD₁₆×C₆₇	—	2	1	no	—
12	#532×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#493×C₆₇	C₂²	1	2	yes	—
16	#486×C₆₇	C₂²	1	2	yes	—
17	#464×C₆₇	—	2	2	no	—
18	#462×C₆₇	C₂²	1	2	yes	complemented
19	#460×C₆₇	—	2	2	no	—
20	#460×C₆₇	C₂²	1	4	yes	—
21	#460×C₆₇	C₂²	1	2	yes	complemented
22	#459×C₆₇	—	2	2	no	—
23	#447×C₆₇	—	2	2	no	—
24	C₈×Q₈×C₆₇	—	2	2	no	—
25	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
26	#431×C₆₇	—	2	2	no	—
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	#423×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#417×C₆₇	—	2	1	no	—
32	C₂×#102×C₆₇	—	2	1	no	—
33	#335×C₆₇	C₂²	1	4	yes	complemented
34	#332×C₆₇	C₂²	1	2	yes	complemented
35	#331×C₆₇	C₂²	1	4	yes	—
36	#331×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#41×C₆₇	—	2	1	no	—
38	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	—	2	1	no	—
41	C₂²×D₄×C₆₇	—	2	1	no	—
42	#137×C₆₇	—	4	4	no	—
43	C₂×Q₁₆×C₆₇	—	2	2	no	—
44	C₂×SD₁₆×C₆₇	—	2	4	no	—
45	C₂×SD₁₆×C₆₇	—	2	2	no	—
46	C₂×SD₁₆×C₆₇	—	4	4	no	—
47	C₂×D₈×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	#127×C₆₇	—	2	1	no	—
52	#127×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#127×C₆₇	C₂³	1	2	yes	—
55	#127×C₆₇	C₂³	1	1	yes	characteristic
56	#124×C₆₇	—	2	1	no	—
57	#121×C₆₇	—	2	1	no	—
58	#121×C₆₇	—	2	1	no	—
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×Q₈×C₆₇	—	2	1	no	—
61	C₄×Q₈×C₆₇	D₄	1	2	yes	—
62	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₇	—	2	4	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	—	2	1	no	—
72	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
73	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	#102×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	2	4	no	—
79	#102×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	—	2	1	no	—
81	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	1	no	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₂⁴×C₆₇	—	4	1	no	—
86	#41×C₆₇	—	4	2	no	—
87	#41×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	2	1	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	D₈	1	4	yes	complemented
92	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	1	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	4	1	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	1	no	—
103	C₂×D₄×C₆₇	—	4	2	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂×D₄×C₆₇	—	8	2	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	4	1	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
113	Q₁₆×C₆₇	—	4	4	no	—
114	SD₁₆×C₆₇	—	4	8	no	—
115	SD₁₆×C₆₇	—	4	4	no	—
116	D₈×C₆₇	—	8	4	no	—
117	M₁₆×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	4	2	no	—
122	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
129	C₂²⋊C₄×C₆₇	—	4	1	no	—
130	C₂²⋊C₄×C₆₇	—	4	1	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	#41	1	1	yes	characteristic
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂³×C₆₇	—	4	1	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	8	2	no	—
140	C₂³×C₆₇	—	8	2	no	—
141	C₂³×C₆₇	—	8	2	no	—
142	C₂³×C₆₇	#121	1	1	yes	characteristic
143	Q₈×C₆₇	—	2	2	no	—
144	Q₈×C₆₇	—	2	4	no	—
145	Q₈×C₆₇	—	4	1	no	—
146	Q₈×C₆₇	—	4	1	no	—
147	Q₈×C₆₇	C₂×D₈	1	4	yes	—
148	D₄×C₆₇	—	4	2	no	—
149	D₄×C₆₇	—	4	1	no	—
150	D₄×C₆₇	—	4	2	no	—
151	D₄×C₆₇	—	4	1	no	—
152	D₄×C₆₇	—	4	2	no	—
153	D₄×C₆₇	—	4	4	no	—
154	D₄×C₆₇	—	8	1	no	—
155	D₄×C₆₇	—	8	2	no	—
156	D₄×C₆₇	—	8	2	no	—
157	D₄×C₆₇	—	8	2	no	—
158	D₄×C₆₇	—	8	2	no	—
159	D₄×C₆₇	—	8	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	8	1	no	—
167	C₂×C₄×C₆₇	—	8	2	no	—
168	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
170	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₆₇	#135	1	2	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
176	C₈×C₆₇	—	2	4	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	8	2	no	—
183	C₂²×C₆₇	—	8	1	no	—
184	C₂²×C₆₇	—	8	1	no	—
185	C₂²×C₆₇	—	8	2	no	—
186	C₂²×C₆₇	—	8	2	no	—
187	C₂²×C₆₇	—	8	1	no	—
188	C₂²×C₆₇	—	16	4	no	—
189	C₂²×C₆₇	#466	1	2	yes	—
190	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	4	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	2	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	8	1	no	—
197	C₄×C₆₇	#466	1	2	yes	—
198	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
200	C₄×C₆₇	#577	1	2	yes	—
201	#19398	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	2	2	no	—
203	C₂×C₆₇	—	16	1	no	—
204	C₂×C₆₇	—	16	2	no	—
205	C₂×C₆₇	#3043	1	1	yes	characteristic
206	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
207	C₂×C₆₇	#3068	1	1	yes	characteristic
208	#3205	C₂×C₆₇	1	1	yes	characteristic
209	#3037	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂×#460	C₂×C₆₇	1	1	yes	characteristic, complemented
211	#2982	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1682	C₂×C₆₇	1	4	yes	complemented
213	#1668	C₂×C₆₇	1	4	yes	complemented
214	#1571	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1545	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1540	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#19398	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×#137	—	2	1	no	—
219	C₂²×SD₁₆	—	2	1	no	—
220	#532	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
223	#493	C₂²×C₆₇	1	2	yes	—
224	#486	C₂²×C₆₇	1	2	yes	—
225	#464	—	2	2	no	—
226	#462	C₂²×C₆₇	1	2	yes	complemented
227	#460	—	2	2	no	—
228	#460	C₂²×C₆₇	1	4	yes	—
229	#460	C₂²×C₆₇	1	2	yes	complemented
230	#459	—	2	2	no	—
231	#447	—	2	2	no	—
232	C₈×Q₈	—	2	2	no	—
233	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
234	#431	—	2	2	no	—
235	#431	C₂²×C₆₇	1	1	yes	characteristic
236	#431	C₂²×C₆₇	1	1	yes	characteristic
237	#423	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
239	#417	—	2	1	no	—
240	C₂×#102	—	2	1	no	—
241	#335	C₂²×C₆₇	1	4	yes	complemented
242	#332	C₂²×C₆₇	1	2	yes	complemented
243	#331	C₂²×C₆₇	1	4	yes	—
244	#331	C₂²×C₆₇	1	2	yes	complemented
245	C₂×#41	—	2	1	no	—
246	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	—	2	1	no	—
249	C₂²×D₄	—	2	1	no	—
250	#137	—	4	4	no	—
251	C₂×Q₁₆	—	2	2	no	—
252	C₂×SD₁₆	—	2	4	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	4	4	no	—
255	C₂×D₈	—	2	4	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂²×C₈	—	2	1	no	—
258	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
259	#127	—	2	1	no	—
260	#127	—	2	1	no	—
261	#127	—	2	1	no	—
262	#127	C₂³×C₆₇	1	2	yes	—
263	#127	C₂³×C₆₇	1	1	yes	characteristic
264	#124	—	2	1	no	—
265	#121	—	2	1	no	—
266	#121	—	2	1	no	—
267	C₄×Q₈	—	2	1	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	D₄×C₆₇	1	2	yes	—
270	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
271	C₄×D₄	—	2	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
277	C₄⋊C₈	—	2	4	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
281	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
282	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
283	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
284	#103	—	2	2	no	—
285	#102	—	2	4	no	—
286	#102	—	2	4	no	—
287	#102	—	2	2	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
290	C₄×C₈	—	2	4	no	—
291	C₄×C₈	—	2	1	no	—
292	C₄×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂⁴	—	4	1	no	—
294	#41	—	4	2	no	—
295	#41	—	4	2	no	—
296	C₂×Q₈	—	2	1	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	1	no	—
299	C₂×Q₈	D₈×C₆₇	1	4	yes	complemented
300	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂×D₄	—	8	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
321	Q₁₆	—	4	4	no	—
322	SD₁₆	—	4	8	no	—
323	SD₁₆	—	4	4	no	—
324	D₈	—	8	4	no	—
325	M₁₆	—	4	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²⋊C₄	—	4	1	no	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₄²	#41×C₆₇	1	1	yes	characteristic
344	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
345	C₂³	—	4	1	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	8	2	no	—
348	C₂³	—	8	2	no	—
349	C₂³	—	8	2	no	—
350	C₂³	#121×C₆₇	1	1	yes	characteristic
351	Q₈	—	2	4	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	4	1	no	—
354	Q₈	—	4	1	no	—
355	Q₈	C₂×D₈×C₆₇	1	4	yes	—
356	D₄	—	4	2	no	—
357	D₄	—	4	1	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	1	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	4	no	—
362	D₄	—	8	2	no	—
363	D₄	—	8	1	no	—
364	D₄	—	8	2	no	—
365	D₄	—	8	2	no	—
366	D₄	—	8	2	no	—
367	D₄	—	8	2	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	8	1	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
377	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
378	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
380	C₂×C₄	#135×C₆₇	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
384	C₈	—	2	4	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	8	2	no	—
391	C₂²	—	8	1	no	—
392	C₂²	—	8	1	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	16	4	no	—
397	C₂²	#466×C₆₇	1	2	yes	—
398	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	4	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	1	no	—
405	C₄	#466×C₆₇	1	2	yes	—
406	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
407	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
408	C₄	#577×C₆₇	1	2	yes	—
409	C₂	—	2	2	no	—
410	C₂	—	16	2	no	—
411	C₂	—	16	1	no	—
412	C₂	C₆₇×#3043	1	1	yes	characteristic
413	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
414	C₂	C₆₇×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3037	C₂	semidirect
3. C₂×#460×C₆₇	C₂	semidirect
4. C₆₇×#2982	C₂	semidirect
5. C₆₇×#1682	C₂	semidirect
6. C₆₇×#1668	C₂	semidirect
7. C₆₇×#1571	C₂	semidirect
8. C₆₇×#1545	C₂	semidirect
9. C₆₇×#1540	C₂	semidirect
18. #462×C₆₇	C₂²	semidirect
21. #460×C₆₇	C₂²	semidirect
25. C₈×Q₈×C₆₇	C₂²	semidirect
33. #335×C₆₇	C₂²	semidirect
34. #332×C₆₇	C₂²	semidirect
36. #331×C₆₇	C₂²	semidirect
91. C₂×Q₈×C₆₇	D₈	semidirect
201. #19398	C₆₇	direct (N × H)
209. #3037	C₂×C₆₇	semidirect
210. C₂×#460	C₂×C₆₇	semidirect
211. #2982	C₂×C₆₇	semidirect
212. #1682	C₂×C₆₇	semidirect
213. #1668	C₂×C₆₇	semidirect
214. #1571	C₂×C₆₇	semidirect
215. #1545	C₂×C₆₇	semidirect
216. #1540	C₂×C₆₇	semidirect
217. C₆₇	#19398	direct (N × H)
226. #462	C₂²×C₆₇	semidirect
229. #460	C₂²×C₆₇	semidirect
233. C₈×Q₈	C₂²×C₆₇	semidirect
241. #335	C₂²×C₆₇	semidirect
242. #332	C₂²×C₆₇	semidirect
244. #331	C₂²×C₆₇	semidirect
299. C₂×Q₈	D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3205	2
C₆₇×#3037	2
C₂×#460×C₆₇	2
C₆₇×#2982	2
C₆₇×#1682	2
C₆₇×#1668	2
C₆₇×#1571	2
C₆₇×#1545	2
C₆₇×#1540	2
#19398	67

Subgroups — C67×#19398 #30698649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19398 #30698649