Subgroups — C₆₇×#17922 #30697173

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2981	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1586	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1581	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1580	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1572	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1484	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#532×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	#480×C₆₇	C₄	1	2	yes	—
15	#474×C₆₇	C₄	1	2	yes	—
16	#459×C₆₇	C₄	1	2	yes	—
17	#447×C₆₇	C₄	1	2	yes	—
18	C₄×D₈×C₆₇	C₂²	1	2	yes	—
19	C₄×D₈×C₆₇	C₂²	1	2	yes	—
20	#436×C₆₇	C₂²	1	2	yes	complemented
21	#435×C₆₇	—	2	2	no	—
22	C₈×D₄×C₆₇	—	2	2	no	—
23	C₈×D₄×C₆₇	C₂²	1	2	yes	—
24	C₈×D₄×C₆₇	C₂²	1	2	yes	—
25	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
26	#432×C₆₇	C₂²	1	1	yes	characteristic
27	#431×C₆₇	C₂²	1	1	yes	characteristic
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	#431×C₆₇	C₂²	1	1	yes	characteristic
30	#408×C₆₇	—	2	2	no	—
31	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	#335×C₆₇	C₂²	1	2	yes	complemented
34	#334×C₆₇	C₂²	1	2	yes	complemented
35	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
36	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
38	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
39	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	#322×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#124×C₆₇	—	2	1	no	—
56	#121×C₆₇	—	2	1	no	—
57	#121×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₄×D₄×C₆₇	—	2	1	no	—
60	C₄×D₄×C₆₇	—	2	1	no	—
61	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#107×C₆₇	C₂×C₄	1	2	yes	—
68	#107×C₆₇	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
70	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
71	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
73	#102×C₆₇	C₂×C₄	1	2	yes	—
74	#102×C₆₇	C₂×C₄	1	2	yes	—
75	#102×C₆₇	C₂×C₄	1	2	yes	—
76	#102×C₆₇	C₂×C₄	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	1	no	—
78	C₂²⋊C₈×C₆₇	—	2	1	no	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	C₂²⋊C₈×C₆₇	—	2	2	no	—
81	C₂²⋊C₈×C₆₇	—	2	2	no	—
82	C₂²⋊C₈×C₆₇	—	2	2	no	—
83	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
84	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
85	#97×C₆₇	—	2	1	no	—
86	#97×C₆₇	—	2	1	no	—
87	C₄×C₈×C₆₇	—	2	1	no	—
88	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂⁴×C₆₇	—	2	1	no	—
95	#41×C₆₇	—	8	2	no	—
96	C₂×Q₈×C₆₇	—	2	1	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	1	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	8	2	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
112	D₈×C₆₇	—	2	4	no	—
113	D₈×C₆₇	—	2	4	no	—
114	M₁₆×C₆₇	—	4	2	no	—
115	M₁₆×C₆₇	—	8	2	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	8	2	no	—
121	C₂×C₈×C₆₇	—	8	2	no	—
122	C₂×C₈×C₆₇	—	8	2	no	—
123	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₂²⋊C₄×C₆₇	—	2	2	no	—
134	C₂²⋊C₄×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	#41	1	1	yes	characteristic
138	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₂³×C₆₇	—	2	4	no	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	8	1	no	—
143	C₂³×C₆₇	—	8	1	no	—
144	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
145	Q₈×C₆₇	—	8	1	no	—
146	D₄×C₆₇	—	2	4	no	—
147	D₄×C₆₇	—	2	2	no	—
148	D₄×C₆₇	—	2	2	no	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	8	1	no	—
151	D₄×C₆₇	—	8	1	no	—
152	D₄×C₆₇	—	8	2	no	—
153	D₄×C₆₇	#131	1	4	yes	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	8	2	no	—
164	C₂×C₄×C₆₇	—	8	1	no	—
165	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₈×C₆₇	—	2	2	no	—
172	C₈×C₆₇	—	2	2	no	—
173	C₈×C₆₇	—	2	2	no	—
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	—	2	2	no	—
176	C₈×C₆₇	—	4	1	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	4	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	8	4	no	—
189	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	C₂×#131	1	2	yes	—
197	C₄×C₆₇	#576	1	2	yes	—
198	#17922	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂×C₆₇	—	4	1	no	—
200	C₂×C₆₇	—	4	4	no	—
201	C₂×C₆₇	—	8	2	no	—
202	C₂×C₆₇	#2929	1	1	yes	characteristic
203	C₂×C₆₇	#2948	1	1	yes	characteristic
204	C₂×C₆₇	#2956	1	1	yes	characteristic
205	#3100	C₂×C₆₇	1	1	yes	characteristic
206	#2981	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#1586	C₂×C₆₇	1	2	yes	complemented
209	#1581	C₂×C₆₇	1	2	yes	complemented
210	#1580	C₂×C₆₇	1	2	yes	complemented
211	C₈×D₈	C₂×C₆₇	1	2	yes	complemented
212	#1572	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#1490	C₂×C₆₇	1	1	yes	characteristic, complemented
214	#1484	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1453	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₆₇	#17922	1	1	yes	characteristic, direct product factor, Sylow
217	#532	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
219	#480	C₄×C₆₇	1	2	yes	—
220	#474	C₄×C₆₇	1	2	yes	—
221	#459	C₄×C₆₇	1	2	yes	—
222	#447	C₄×C₆₇	1	2	yes	—
223	C₄×D₈	C₂²×C₆₇	1	2	yes	—
224	C₄×D₈	C₂²×C₆₇	1	2	yes	—
225	#436	C₂²×C₆₇	1	2	yes	complemented
226	#435	—	2	2	no	—
227	C₈×D₄	—	2	2	no	—
228	C₈×D₄	C₂²×C₆₇	1	2	yes	—
229	C₈×D₄	C₂²×C₆₇	1	2	yes	—
230	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
231	#432	C₂²×C₆₇	1	1	yes	characteristic
232	#431	C₂²×C₆₇	1	1	yes	characteristic
233	#431	C₂²×C₆₇	1	1	yes	characteristic
234	#431	C₂²×C₆₇	1	1	yes	characteristic
235	#408	—	2	2	no	—
236	C₂×C₂²⋊C₈	—	2	2	no	—
237	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
238	#335	C₂²×C₆₇	1	2	yes	complemented
239	#334	C₂²×C₆₇	1	2	yes	complemented
240	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
241	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
242	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
243	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
244	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
245	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
246	#322	C₂²×C₆₇	1	1	yes	characteristic, complemented
247	C₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
248	C₂×#41	—	2	1	no	—
249	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
252	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×M₁₆	—	2	1	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	—	2	2	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
259	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
260	#124	—	2	1	no	—
261	#121	—	2	1	no	—
262	#121	C₂×C₄×C₆₇	1	2	yes	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
267	C₄×D₄	C₂³×C₆₇	1	2	yes	—
268	C₄×D₄	C₂³×C₆₇	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
270	C₂×C₂²⋊C₄	—	2	1	no	—
271	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
272	#107	C₂×C₄×C₆₇	1	2	yes	—
273	#107	C₂×C₄×C₆₇	1	2	yes	—
274	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
275	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
276	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
277	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
278	#102	C₂×C₄×C₆₇	1	2	yes	—
279	#102	C₂×C₄×C₆₇	1	2	yes	—
280	#102	C₂×C₄×C₆₇	1	2	yes	—
281	#102	C₂×C₄×C₆₇	1	2	yes	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	—	2	2	no	—
287	C₂²⋊C₈	—	2	2	no	—
288	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
289	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
290	#97	—	2	1	no	—
291	#97	—	2	1	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
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	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂⁴	—	2	1	no	—
300	#41	—	8	2	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	8	2	no	—
314	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
317	D₈	—	2	4	no	—
318	D₈	—	2	4	no	—
319	M₁₆	—	4	2	no	—
320	M₁₆	—	8	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	8	2	no	—
326	C₂×C₈	—	8	2	no	—
327	C₂×C₈	—	8	2	no	—
328	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
330	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
335	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
341	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
342	C₄²	#41×C₆₇	1	1	yes	characteristic
343	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
344	C₂³	—	2	4	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	8	1	no	—
348	C₂³	—	8	1	no	—
349	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
350	Q₈	—	8	1	no	—
351	D₄	—	2	4	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	2	2	no	—
355	D₄	—	8	1	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	2	no	—
358	D₄	#131×C₆₇	1	4	yes	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	1	no	—
370	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
376	C₈	—	2	2	no	—
377	C₈	—	2	2	no	—
378	C₈	—	2	2	no	—
379	C₈	—	2	2	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	8	4	no	—
394	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	C₂×#131×C₆₇	1	2	yes	—
402	C₄	#576×C₆₇	1	2	yes	—
403	C₂	—	4	1	no	—
404	C₂	—	4	4	no	—
405	C₂	—	8	2	no	—
406	C₂	C₆₇×#2929	1	1	yes	characteristic
407	C₂	C₆₇×#2948	1	1	yes	characteristic
408	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3100	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2981	C₂	semidirect
3. C₆₇×#2972	C₂	semidirect
4. C₆₇×#1586	C₂	semidirect
5. C₆₇×#1581	C₂	semidirect
6. C₆₇×#1580	C₂	semidirect
7. C₈×D₈×C₆₇	C₂	semidirect
8. C₆₇×#1572	C₂	semidirect
9. C₆₇×#1490	C₂	semidirect
10. C₆₇×#1484	C₂	semidirect
11. C₆₇×#1453	C₂	semidirect
20. #436×C₆₇	C₂²	semidirect
25. C₈×D₄×C₆₇	C₂²	semidirect
33. #335×C₆₇	C₂²	semidirect
34. #334×C₆₇	C₂²	semidirect
37. D₄⋊C₈×C₆₇	C₂²	semidirect
38. D₄⋊C₈×C₆₇	C₂²	semidirect
39. #322×C₆₇	C₂²	semidirect
40. #322×C₆₇	C₂²	semidirect
41. #322×C₆₇	C₂²	semidirect
42. C₈²×C₆₇	C₂²	semidirect
198. #17922	C₆₇	direct (N × H)
206. #2981	C₂×C₆₇	semidirect
207. #2972	C₂×C₆₇	semidirect
208. #1586	C₂×C₆₇	semidirect
209. #1581	C₂×C₆₇	semidirect
210. #1580	C₂×C₆₇	semidirect
211. C₈×D₈	C₂×C₆₇	semidirect
212. #1572	C₂×C₆₇	semidirect
213. #1490	C₂×C₆₇	semidirect
214. #1484	C₂×C₆₇	semidirect
215. #1453	C₂×C₆₇	semidirect
216. C₆₇	#17922	direct (N × H)
225. #436	C₂²×C₆₇	semidirect
230. C₈×D₄	C₂²×C₆₇	semidirect
238. #335	C₂²×C₆₇	semidirect
239. #334	C₂²×C₆₇	semidirect
242. D₄⋊C₈	C₂²×C₆₇	semidirect
243. D₄⋊C₈	C₂²×C₆₇	semidirect
244. #322	C₂²×C₆₇	semidirect
245. #322	C₂²×C₆₇	semidirect
246. #322	C₂²×C₆₇	semidirect
247. C₈²	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3100	2
C₆₇×#2981	2
C₆₇×#2972	2
C₆₇×#1586	2
C₆₇×#1581	2
C₆₇×#1580	2
C₈×D₈×C₆₇	2
C₆₇×#1572	2
C₆₇×#1490	2
C₆₇×#1484	2
C₆₇×#1453	2
#17922	67

Subgroups — C67×#17922 #30697173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17922 #30697173