Subgroups — C₂×C₆₇×#1868 #30700142

Order 17152.

1342 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#410×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1868	C₂	1	8	yes	maximal, direct product factor
6	C₂×#344×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	#568×C₆₇	—	2	2	no	—
9	#568×C₆₇	—	2	4	no	—
10	#568×C₆₇	C₂²	1	4	yes	—
11	C₂×#131×C₆₇	—	2	4	no	—
12	C₂²×M₁₆×C₆₇	—	2	2	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	#424×C₆₇	—	2	4	no	—
18	#424×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
20	#410×C₆₇	C₄	1	4	yes	—
21	#410×C₆₇	C₂²	1	4	yes	—
22	C₂×#99×C₆₇	C₄	1	4	yes	—
23	#408×C₆₇	—	2	2	no	—
24	#408×C₆₇	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
26	#344×C₆₇	C₂²	1	8	yes	complemented
27	C₂×#41×C₆₇	—	2	2	no	—
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	2	no	—
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×#41×C₆₇	D₄	1	2	yes	—
33	C₂×#41×C₆₇	Q₈	1	2	yes	—
34	C₂×#41×C₆₇	C₂³	1	2	yes	—
35	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
36	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	#131×C₆₇	—	2	16	no	—
41	#131×C₆₇	—	2	16	no	—
42	C₂×M₁₆×C₆₇	—	2	4	no	—
43	C₂×M₁₆×C₆₇	—	2	8	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	8	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	—	2	1	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
54	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₂²×C₈×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	—	2	1	no	—
61	C₂²×C₈×C₆₇	D₄	1	4	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	#117×C₆₇	—	2	2	no	—
64	#117×C₆₇	—	2	2	no	—
65	#117×C₆₇	C₂×C₄	1	4	yes	—
66	#117×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	16	no	—
71	#99×C₆₇	C₂×C₄	1	16	yes	—
72	C₂²⋊C₈×C₆₇	—	2	8	no	—
73	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
74	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	#41×C₆₇	—	2	4	no	—
76	#41×C₆₇	—	2	4	no	—
77	#41×C₆₇	—	2	4	no	—
78	#41×C₆₇	—	2	4	no	—
79	#41×C₆₇	—	2	4	no	—
80	#41×C₆₇	—	2	4	no	—
81	#41×C₆₇	—	2	4	no	—
82	#41×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
86	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
92	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
94	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	#41	1	2	yes	—
111	C₂²×C₄×C₆₇	#41	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₆₇	—	2	16	no	—
114	M₁₆×C₆₇	—	2	4	no	—
115	M₁₆×C₆₇	—	2	4	no	—
116	M₁₆×C₆₇	—	2	4	no	—
117	M₁₆×C₆₇	—	2	4	no	—
118	M₁₆×C₆₇	—	2	4	no	—
119	M₁₆×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	—	2	16	no	—
121	C₂×C₈×C₆₇	—	2	4	no	—
122	C₂×C₈×C₆₇	—	2	4	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	4	2	no	—
127	C₂×C₈×C₆₇	—	4	1	no	—
128	C₂×C₈×C₆₇	—	4	1	no	—
129	C₂×C₈×C₆₇	—	4	1	no	—
130	C₂×C₈×C₆₇	—	4	1	no	—
131	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₂²⋊C₄×C₆₇	—	2	4	no	—
135	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
136	C₄²×C₆₇	—	2	4	no	—
137	C₄²×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	—	2	2	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	4	no	—
142	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
144	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂³×C₆₇	C₄×D₄	1	2	yes	—
146	C₂³×C₆₇	C₄×D₄	1	2	yes	—
147	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
148	Q₈×C₆₇	—	2	4	no	—
149	Q₈×C₆₇	—	2	4	no	—
150	D₄×C₆₇	—	2	8	no	—
151	D₄×C₆₇	—	2	8	no	—
152	D₄×C₆₇	—	2	4	no	—
153	D₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	4	8	no	—
162	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₆₇	#120	1	2	yes	—
168	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
169	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
175	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
176	C₂×C₄×C₆₇	#123	1	2	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
181	C₈×C₆₇	—	2	8	no	—
182	C₈×C₆₇	—	4	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	1	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	#385	1	1	yes	characteristic
194	C₂²×C₆₇	#385	1	2	yes	—
195	C₂²×C₆₇	#385	1	1	yes	characteristic
196	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
198	C₄×C₆₇	—	2	4	no	—
199	C₄×C₆₇	—	4	8	no	—
200	C₄×C₆₇	#385	1	1	yes	characteristic
201	C₄×C₆₇	#385	1	1	yes	characteristic
202	C₄×C₆₇	#385	1	2	yes	—
203	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
204	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
205	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
206	C₂×#1868	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₆₇	—	2	2	no	—
208	C₂×C₆₇	—	2	1	no	—
209	C₂×C₆₇	—	2	1	no	—
210	C₂×C₆₇	—	2	2	no	—
211	C₂×C₆₇	—	4	4	no	—
212	C₂×C₆₇	#1868	1	2	yes	direct product factor
213	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
214	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
215	C₂×#424	C₂×C₆₇	1	2	yes	complemented
216	C₂×#410	C₂×C₆₇	1	1	yes	characteristic
217	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
218	#1868	C₂×C₆₇	1	8	yes	direct product factor
219	C₂×#344	C₂×C₆₇	1	2	yes	complemented
220	C₆₇	C₂×#1868	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
222	#568	—	2	2	no	—
223	#568	—	2	4	no	—
224	#568	C₂²×C₆₇	1	4	yes	—
225	C₂×#131	—	2	4	no	—
226	C₂²×M₁₆	—	2	2	no	—
227	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
228	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
229	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×#117	C₂²×C₆₇	1	2	yes	—
231	#424	—	2	4	no	—
232	#424	C₂²×C₆₇	1	8	yes	complemented
233	C₂×C₄⋊C₈	—	2	4	no	—
234	#410	C₄×C₆₇	1	4	yes	—
235	#410	C₂²×C₆₇	1	4	yes	—
236	C₂×#99	C₄×C₆₇	1	4	yes	—
237	#408	—	2	2	no	—
238	#408	C₂²×C₆₇	1	4	yes	—
239	C₂×C₂²⋊C₈	—	2	2	no	—
240	#344	C₂²×C₆₇	1	8	yes	complemented
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	2	2	no	—
243	C₂×#41	—	2	2	no	—
244	C₂×#41	—	2	2	no	—
245	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
246	C₂×#41	D₄×C₆₇	1	2	yes	—
247	C₂×#41	Q₈×C₆₇	1	2	yes	—
248	C₂×#41	C₂³×C₆₇	1	2	yes	—
249	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
250	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
251	C₂²×D₄	Q₈×C₆₇	1	1	yes	characteristic
252	C₂³×C₄	D₄×C₆₇	1	2	yes	—
253	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
254	#131	—	2	16	no	—
255	#131	—	2	16	no	—
256	C₂×M₁₆	—	2	4	no	—
257	C₂×M₁₆	—	2	8	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	8	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
267	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
268	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
269	C₂²×C₈	—	2	4	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	2	no	—
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	—	2	1	no	—
275	C₂²×C₈	D₄×C₆₇	1	4	yes	—
276	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
277	#117	—	2	2	no	—
278	#117	—	2	2	no	—
279	#117	C₂×C₄×C₆₇	1	4	yes	—
280	#117	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
283	C₂×C₄²	—	2	2	no	—
284	C₄⋊C₈	—	2	16	no	—
285	#99	C₂×C₄×C₆₇	1	16	yes	—
286	C₂²⋊C₈	—	2	8	no	—
287	C₂⁴	C₄⋊C₄×C₆₇	1	2	yes	—
288	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
289	#41	—	2	4	no	—
290	#41	—	2	4	no	—
291	#41	—	2	4	no	—
292	#41	—	2	4	no	—
293	#41	—	2	4	no	—
294	#41	—	2	4	no	—
295	#41	—	2	4	no	—
296	#41	—	2	4	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
300	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
306	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
307	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
308	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
315	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
316	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	#41×C₆₇	1	2	yes	—
325	C₂²×C₄	#41×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
327	M₁₆	—	2	16	no	—
328	M₁₆	—	2	4	no	—
329	M₁₆	—	2	4	no	—
330	M₁₆	—	2	4	no	—
331	M₁₆	—	2	4	no	—
332	M₁₆	—	2	4	no	—
333	M₁₆	—	4	2	no	—
334	C₂×C₈	—	2	16	no	—
335	C₂×C₈	—	2	4	no	—
336	C₂×C₈	—	2	4	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	4	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	4	1	no	—
345	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₂²⋊C₄	—	2	4	no	—
349	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
350	C₄²	—	2	4	no	—
351	C₄²	—	2	4	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	4	no	—
356	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
357	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
358	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
359	C₂³	C₄×D₄×C₆₇	1	2	yes	—
360	C₂³	C₄×D₄×C₆₇	1	2	yes	—
361	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	2	4	no	—
364	D₄	—	2	8	no	—
365	D₄	—	2	8	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
378	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
380	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
381	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
382	C₂×C₄	#120×C₆₇	1	2	yes	—
383	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
385	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
386	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
387	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
388	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
389	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
390	C₂×C₄	#123×C₆₇	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
395	C₈	—	2	8	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₂²	—	2	4	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	4	4	no	—
405	C₂²	—	4	4	no	—
406	C₂²	—	8	2	no	—
407	C₂²	#385×C₆₇	1	1	yes	characteristic
408	C₂²	#385×C₆₇	1	2	yes	—
409	C₂²	#385×C₆₇	1	1	yes	characteristic
410	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
411	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
412	C₄	—	2	4	no	—
413	C₄	—	4	8	no	—
414	C₄	#385×C₆₇	1	1	yes	characteristic
415	C₄	#385×C₆₇	1	1	yes	characteristic
416	C₄	#385×C₆₇	1	2	yes	—
417	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
418	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
419	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
420	C₂	—	2	2	no	—
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	—	2	2	no	—
424	C₂	—	4	4	no	—
425	C₂	C₆₇×#1868	1	2	yes	direct product factor
426	C₂	C₂×#385×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#120
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	#385
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#424×C₆₇	C₂	semidirect
5. C₆₇×#1868	C₂	direct (N × H)
6. C₂×#344×C₆₇	C₂	semidirect
18. #424×C₆₇	C₂²	semidirect
26. #344×C₆₇	C₂²	semidirect
206. C₂×#1868	C₆₇	direct (N × H)
212. C₂×C₆₇	#1868	direct (N × H)
215. C₂×#424	C₂×C₆₇	semidirect
218. #1868	C₂×C₆₇	direct (N × H)
219. C₂×#344	C₂×C₆₇	semidirect
220. C₆₇	C₂×#1868	direct (N × H)
232. #424	C₂²×C₆₇	semidirect
240. #344	C₂²×C₆₇	semidirect
425. C₂	C₆₇×#1868	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#568×C₆₇	2
C₂×#424×C₆₇	2
C₂×#410×C₆₇	2
C₂×#408×C₆₇	2
C₆₇×#1868	2
C₂×#344×C₆₇	2
C₂×#1868	67

Subgroups — C2×C67×#1868 #30700142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1868 #30700142